The cipher suite Ascon v1.2 already provides authenticated encryption schemes, hash, and extendable output functions. Furthermore, the underlying permutation is also used in two instances of Isap v2.0, an authenticated encryption scheme designed to provide enhanced robustness against side-channel and fault attacks. In this paper, we enrich the functionality one can get out of Ascon's permutation by providing efficient Pseudorandom Functions (PRFs), a Message Authentication Code (MAC) and a fast short-input PRF for messages up to 128 bits.

This note shows practical committing attacks against Rocca-S, an authenticated encryption with associated data scheme designed for 6G applications. Previously, the best complexity of the attack was $2^{64}$ by Derbez et al. in ToSC 2024(1)/FSE 2024. We show that the committing attack against Rocca by Takeuchi et al. in ToSC 2024(2)/FSE 2025 can be applied to Rocca-S, where Rocca is an earlier version of Rocca-S. We show a concrete test vector of our attack. We also point out a committing attack that exploits equivalent keys.

The Ascon specification defines among others an encryption scheme offering authenticated encryption with associated data (AEAD) which is based on a duplex mode of a sponge. With that it is the first of such algorithm selected and about to be standardized by NIST. The sponge size is comparatively small, 320 bits, as expected for lightweight cryptography. With that, the strength of the defined AEAD algorithm is limited to 128 bits. Albeit, the definition of the Ascon AEAD algorithm integrates with the associated sponge, it is mathematically not bound to exactly this sponge function. Thus, the Ascon AEAD specification can be used with a different sponge and still operate as defined by the Ascon authors. This specification defines the Ascon-Keccak AEAD algorithm which replaces the Ascon sponge with the Keccak sponge, leaving the Ascon AEAD algorithm unchanged. The selected parameters for Ascon-Keccak AEAD offer two algorithm strengths: Ascon-Keccak 256 with a classic security strength of 256 bits and a quantum security strength of 128 bits. In addition, Ascon-Keccak 512 provides an algorithm with 512 bit classic security strength and 256 bit quantum security strength. The selected parameters for Ascon-Keccak 256 offer a significant higher performance on 64-bit architectures than Ascon-128 and Ascon-128a. The performance of Ascon-Keccak 512 is in league with Ascon-128. Yet, with the Keccak sponge size of 1600 bits, Ascon-Keccak cannot be considered a lightweight cryptographic algorithm any more. A reference implementation of the algorithm is provided as referenced in the document.

XS-circuits describe cryptographic primitives that utilize 2 operations on binary words of fixed length: X) bitwise modulo 2 addition and S) substitution. The words are interpreted as elements of a field of characteristic 2. In this paper, we develop a model of XS-circuits according to which several instances of a simple round circuit containing only one S operation are linked together and form a compound circuit called a cascade. S operations of a cascade are interpreted as independent round oracles. When a cascade processes a pair of different inputs, some round oracles get different queries, these oracles are activated. The more activations, the higher security guarantees against differential cryptanalysis the cascade provides. We introduce the notion of the guaranteed number of activations, that is, the minimum number of activations over all choices of the base field, round oracles and pairs of inputs. We show that the guaranteed number of activations is related to the minimum distance of the linear code associated with the cascade. It is also related to the minimum number of occurrences of units in segments of binary linear recurrence sequences whose characteristic polynomial is determined by the round circuit. We provide an algorithm for calculating the guaranteed number of activations. We show how to use the algorithm to deal with linear activations related to linear cryptanalysis.

The best-known distinguisher on 7-round Ascon-128 and Ascon-128a AEAD uses a 60-dimensional cube where the nonce bits are set to be equal in the third and fourth rows of the Ascon state during initialization (Rohit et al. ToSC 2021/1). It was not known how to use this distinguisher to mount key-recovery attacks. In this paper, we investigate this problem using a new strategy called \textit{break-fix} for the conditional cube attack. The idea is to introduce slightly-modified cubes which increase the degrees of 7-round output bits to be more than 59 (break phase) and then find key conditions which can bring the degree back to 59 (fix phase). Using this idea, key-recovery attacks on 7-round Ascon-128, Ascon-128a and Ascon-80pq are proposed. The attacks have better time/memory complexities than the existing attacks, and in some cases improve the weak-key attacks as well.

A growing number of lightweight block ciphers are proposed for environments such as the Internet of Things. An important contribution to the reduced implementation cost is a block length n of 64 or 96 bits rather than 128 bits. As a consequence, encryption modes and message authentication code (MAC) algorithms require security beyond the 2^{n/2} birthday bound. This paper provides an extensive treatment of MAC algorithms that offer beyond birthday bound PRF security for both nonce-respecting and nonce-misusing adversaries. We study constructions that use two block cipher calls, one universal hash function call and an arbitrary number of XOR operations. We start with the separate problem of generically identifying all possible secure n-to-n-bit pseudorandom functions (PRFs) based on two block cipher calls. The analysis shows that the existing constructions EDM, SoP, and EDMD are the only constructions of this kind that achieve beyond birthday bound security. Subsequently we deliver an exhaustive treatment of MAC algorithms, where the outcome of a universal hash function evaluation on the message may be entered at any point in the computation of the PRF. We conclude that there are a total amount of nine schemes that achieve beyond birthday bound security, and a tenth construction that cannot be proven using currently known proof techniques. For these former nine MAC algorithms, three constructions achieve optimal n-bit security in the nonce-respecting setting, but are completely insecure if the nonce is reused. The remaining six constructions have 3n/4-bit security in the nonce-respecting setting, and only four out of these six constructions still achieve beyond the birthday bound security in the case of nonce misuse.

This paper's purpose is to give a new method of analyzing Beale Cipher 1 and Cipher 3 and to show that there is no key which will decipher them into sentences. Previous research has largely used statistical methods to either decipher them or prove they have no solution. Some of these methods show that there is a high probability, but not certainty that they are unsolvable. Both ciphers remain unsolved. The methods used in this paper are not statistical ones based on thousands of samples. The evidence given here shows there is a high correlation between locations of certain numbers in the ciphers with locations in the written text that was given with these ciphers in the 1885 pamphlet called "The Beale Papers". Evidence is correlated with a long monotonically increasing Gillogly String in Cipher 1, when translated with the Declaration of Independence given in the pamphlet. The Beale Papers' writer was anonymous, and words in the three written letters in the 1885 pamphlet are compared with locations of numbers in the ciphers to show who the writer was. Emphasis is on numbers which are controllable by the encipherer. Letter location sums are used when they are the most plausible ones found. Evidence supports the statement that Cipher 1 and Cipher 3 are unintelligible. It also supports the statement that they were designed to have no intelligible sentences because they were part of a complex game made by the anonymous writer of The Beale Papers.

he unique design of the FLIP cipher necessitated a generalization of standard cryptographic criteria for Boolean functions used in stream ciphers, prompting a focus on properties specific to subsets of $\mathbb{F}_2^n$ rather than the entire set. This led to heightened interest in properties related to fixed Hamming weight sets and the corresponding partition of $\mathbb{F}_2^n$ into n+1 such sets. Consequently, the concept of Weightwise Almost Perfectly Balanced (WAPB) functions emerged, which are balanced on each of these sets.Various studies have since proposed WAPB constructions and examined their cryptographic parameters for use in stream cipher filters. In this article, we introduce a general approach to constructing WAPB functions using the concept of order, which simplifies implementation and enhances cryptographic strength. We present two new constructions: a recursive method employing multiple orders on binary strings, and another utilizing just two orders. We establish lower bounds for nonlinearity and weightwise nonlinearities within these classes. By instantiating specific orders, we demonstrate that some achieve minimal algebraic immunity, while others provide functions with guaranteed optimal algebraic immunity. Experimental results in 8 and 16 variables indicate that using orders based on field representation significantly outperforms other methods in terms of both global and weightwise algebraic immunity and nonlinearity. Additionally, we extend the recursive construction to create WAPB functions for any value of n, with experiments in 10, 12, and 14 variables confirming that these order-based functions exhibit robust cryptographic parameters. In particular, those based on field orders display optimal degrees and algebraic immunity, and strong weightwise nonlinearities and algebraic immunities.

Tweakable HCTR is an tweakable enciphering proposed by Dutta and Nandi in Indocrypt 2018. It provides beyond birthday bound security when each tweak value is not used too frequently. More importantly for this note, its security bound degrades linearly with the maximum input length. We show in this note that this is not true by showing a single query distinguisher with advantage $O(l^2/2^n)$ where $l$ is the length of that query. The distinguisher does not break the beyond-birthday-bound claim but gives higher advantage than the claimed bound.

Liskov, Rivest and Wagner laid the theoretical foundations for tweakable block ciphers (TBC). In a seminal paper, they proposed two (up to) birthday-bound secure design strategies --- LRW1 and LRW2 --- to convert any block cipher into a TBC. Several of the follow-up works consider cascading of LRW-type TBCs to construct beyond-the-birthday bound (BBB) secure TBCs. Landecker et al. demonstrated that just two-round cascading of LRW2 can already give a BBB security. Bao et al. undertook a similar exercise in context of LRW1 with TNT --- a three-round cascading of LRW1 --- that has been shown to achieve BBB security as well. In this paper, we present a CCA distinguisher on TNT that achieves a non-negligible advantage with $ O(2^{n/2}) $ queries, directly contradicting the security claims made by the designers. We provide a rigorous and complete advantage calculation coupled with experimental verification that further support our claim. Next, we provide new and simple proofs of birthday-bound CCA security for both TNT and its single-key variant, which confirm the tightness of our attack. Furthering on to a more positive note, we show that adding just one more block cipher call, referred as 4-LRW1, does not just re-establish the BBB security, but also amplifies it up to $ 2^{3n/4} $ queries. As a side-effect of this endeavour, we propose a new abstraction of the cascaded LRW-design philosophy, referred to as the LRW+ paradigm, comprising two block cipher calls sandwiched between a pair of tweakable universal hashes. This helps us to provide a modular proof covering all cascaded LRW constructions with at least $ 2 $ rounds, including 4-LRW1, and its more established relative, the well-known CLRW2, or more aptly, 2-LRW2.

WESP is a new encryption algorithm that is based on equation systems, in which the equations are generated using the values of tables that act as the encryption key, and the equations having features making them suitable for cryptographic use. The algorithm is defined, and its properties are discussed. Besides just describing the algorithm, also reasons are presented why the algorithm works the way it works. The key size in WESP can be altered and has no upper limit, and typically the key size is bigger than currently commonly used keys. A calculation is presented, calculating how many bytes can be securely encrypted before the algorithm might start to repeat it’s sequence of encrypting bytes, and that this period can be adjusted to be arbitrarily large. It is also shown that withing the period the resulting stream of encrypting bytes is statistically uniformly distributed. It is also shown that if the encryption tables are not known, the equations in the system of equations cannot be known, and it is demonstrated that the system of equations cannot be solved if the equations are not known, and thus the encryption cannot be broken in a closed form. Then, we calculate for all symbols used in the algorithm, the minimum amount of trials needed, in order to be able to verity the trials. Since the algorithm is constantly updating key values, verification becomes impossible if equations are not evaluated in order. The calculation shows that the minimum number of trials required is such that the number of trials, i.e., the time required to break the encryption, increases exponentially as the key size grows. Since there is no upper limit on the key size there is neither any upper limit on the time it requires to break the encryption.

We have investigated both the padding scheme and the applicability of algebraic attacks to both XHash8 and XHash12. The only vulnerability of the padding scheme we can find is plausibly applicable only in the multi-rate setting---for which the authors make no claim---and is safe otherwise. For algebraic attack relying on the computation and exploitation of a Gröbner basis, our survey of the literature suggests to base a security argument on the complexity of the variable elimination step rather than that of the computation of the Gröbner basis itself. Indeed, it turns out that the latter complexity is hard to estimate---and is sometimes litteraly non-existent. Focusing on the elimination step, we propose a generalization of the "FreeLunch" approach which, under a reasonable conjecture about the behaviour of the degree of polynomial ideals of dimension 0, is sufficient for us to argue that both XHash8 and XHash12 are safe against such attacks. We implemented a simplified version of the generation (and resolution) of the corresponding set of equations in SAGE, which allowed us to validate our conjecture at least experimentally, and in fact to show that the lower bound it provides on the ideal degree is not tight---meaning we are a priori understimating the security of these permutations against the algebraic attacks we consider. At this stage, if used as specified, these hash functions seem safe from Gröbner bases-based algebraic attacks.

The Ascon cipher suite has recently become the preferred standard in the NIST Lightweight Cryptography standardization process. Despite its prominence, the initial dedicated security analysis for the Ascon mode was conducted quite recently. This analysis demonstrated that the Ascon AEAD mode offers superior security compared to the generic Duplex mode, but it was limited to a specific scenario: single-user nonce-respecting, with a capacity strictly larger than the key size. In this paper, we eliminate these constraints and provide a comprehensive security analysis of the Ascon AEAD mode in the multi-user setting, where the capacity need not be larger than the key size. Regarding data complexity $D$ and time complexity $T$, our analysis reveals that Ascon achieves AEAD security when $T$ is bounded by $\min\{2^{\kappa}/\mu, 2^c\}$ (where $\kappa$ is the key size, and $\mu$ is the number of users), and $DT$ is limited to $2^b$ (with $b$ denoting the size of the underlying permutation, set at 320 for Ascon). Our results align with NIST requirements, showing that Ascon allows for a tag size as small as 64 bits while supporting a higher rate of 192 bits, provided the number of users remains within recommended limits. However, this security becomes compromised as the number of users increases significantly. To address this issue, we propose a variant of the Ascon mode called LK-Ascon, which enables doubling the key size. This adjustment allows for a greater number of users without sacrificing security, while possibly offering additional resilience against quantum key recovery attacks. We establish tight bounds for LK-Ascon, and furthermore show that both Ascon and LK-Ascon maintain authenticity security even when facing nonce-misuse adversaries.

The XOR of two independent permutations (XoP) is a well-known construction for achieving security beyond the birthday bound when implementing a pseudorandom function using a block cipher (i.e., a pseudorandom permutation). The idealized construction (where the permutations are uniformly chosen and independent) and its variants have been extensively analyzed over nearly 25 years. The best-known asymptotic information-theoretic indistinguishability bound for the XoP construction is $O(q/2^{1.5n})$, derived by Eberhard in 2017, where $q$ is the number of queries and $n$ is the block length. A generalization of the XoP construction outputs the XOR of $r \geq 2$ independent permutations, and has also received significant attention in both the single-user and multi-user settings. In particular, for $r = 3$, the best-known bound (obtained by Choi et al. [ASIACRYPT'22]) is about $q^2/2^{2.5 n}$ in the single-user setting and $\sqrt{u} q_{\max}^2/2^{2.5 n}$ in the multi-user setting (where $u$ is the number of users and $q_{\max}$ is the number of queries per user). In this paper, we prove an indistinguishability bound of $q/2^{(r - 0.5)n}$ for the (generalized) XoP construction in the single-user setting, and a bound of $\sqrt{u} q_{\max}/2^{(r - 0.5)n}$ in the multi-user setting. In particular, for $r=2$, we obtain the bounds $q/2^{1.5n}$ and $\sqrt{u} q_{\max}/2^{1.5n}$ in single-user and multi-user settings, respectively. For $r=3$ the corresponding bounds are $q/2^{2.5n}$ and $\sqrt{u} q_{\max}/2^{2.5n}$. All of these bounds hold assuming $q < 2^{n}/2$ (or $q_{\max} < 2^{n}/2$). Compared to previous works, we improve all the best-known bounds for the (generalized) XoP construction in the multi-user setting, and the best-known bounds for the generalized XoP construction for $r \geq 3$ in the single-user setting (assuming $q \geq 2^{n/2}$). For the basic two-permutation XoP construction in the single-user setting, our concrete bound of $q/2^{1.5n}$ stands in contrast to the asymptotic bound of $O(q/2^{1.5n})$ by Eberhard. Since all of our bounds are matched (up to constant factors) for $q \geq 2^{n/2}$ by attacks published by Patarin in 2008 (and their generalizations to the multi-user setting), they are all tight. We obtain our results by Fourier analysis of Boolean functions. Most of our technical work involves bounding (sums of) Fourier coefficients of the density function associated with sampling without replacement. While the proof of Eberhard relies on similar bounds, our proof is elementary and simpler.

We present a construction, called Kirby, for building a variable-input-length pseudorandom function (VIL-PRF) from a $b$-bit permutation. For this construction we prove a tight bound of $b/2$ bits of security on the PRF distinguishing advantage in the random permutation model and in the multi-user setting. Similar to full-state keyed sponge/duplex, it supports full-state absorbing and additionally supports full-state squeezing, while the sponge/duplex can squeeze at most $b-c$ bits per permutation call, for a security level of $c$ bits. This advantage is especially relevant on constrained platforms when using a permutation with small width $b$. For instance, for $b=256$ at equal security strength the squeezing rate of Kirby is twice that of keyed sponge/duplex. This construction could be seen as a generalization of the construction underlying the stream cipher family Salsa. Furthermore, we define a simple mode on top of Kirby that turns it into a deck function with parallel expansion. This is similar to Farfalle but it has a much smaller memory footprint. Furthermore we prove that in the Kirby construction, the leakage of intermediate states does not allow recovering the key or earlier states.

Truncated differential cryptanalyses were introduced by Knudsen in 1994. They are a well-known family of attacks that has arguably received less attention than some other variants of differential attacks. This paper gives some new insights into the theory of truncated differential attacks, specifically the provable security of SPN ciphers with MDS diffusion matrices against this type of attack. Furthermore, our study extends to various versions within the QARMA family of block ciphers, unveiling the only valid instances of single-tweak attacks on 10-round QARMAv1-64, 10-round QARMAv1-128, and 10- and 11-round QARMAv2-64. These attacks benefit from the optimal truncated differential distinguishers as well as some evolved key-recovery techniques.

We describe a small tweak to Cuckoo filters that allows securing them under insertions using the techniques from Filić et al. (ACM CCS 2022), without the need for an outer PRF call.

The size of the authentication tag represents a significant overhead for applications that are limited by bandwidth or memory. Hence, some authenticated encryption designs have a smaller tag than the required privacy level, which was also suggested by the NIST lightweight cryptography standardization project. In the ToSC 2022, two papers have raised questions about the IND-CCA security of AEAD schemes in this situation. These papers show that (a) online AE cannot provide IND-CCA security beyond the tag length, and (b) it is possible to have IND-CCA security beyond the tag length in a restricted Encrypt-then-Encipher framework. In this paper, we address some of the remaining gaps in this area. Our main result is to show that, for a fixed stretch, Pseudo-Random Injection security implies IND-CCA security as long as the minimum ciphertext size is at least as large as the required IND-CCA security level. We also show that this bound is tight and that any AEAD scheme that allows empty plaintexts with a fixed stretch cannot achieve IND-CCA security beyond the tag length. Next, we look at the weaker notion of MRAE security, and show that two-pass schemes that achieve MRAE security do not achieve IND-CCA security beyond the tag size. This includes SIV and rugged PRPs.

The Advanced Encryption Standard (AES) is one of the most commonly used and analyzed encryption algorithms. In this work, we present new combinations of some prominent attacks on AES, achieving new records in data requirements among attacks, utilizing only $2^4$ and $2^{16}$ chosen plaintexts (CP) for 6-round and 7-round AES-192/256 respectively. One of our attacks is a combination of a meet-in-the-middle (MiTM) attack with a square attack mounted on 6-round AES-192/256 while another attack combines an MiTM attack and an integral attack, utilizing key space partitioning technique, on 7-round AES-192/256. Moreover, we illustrate that impossible differential (ID) attacks can be viewed as the dual of MiTM attacks in certain aspects which enables us to recover the correct key using the meet-in-the-middle (MiTM) technique instead of sieving through all potential wrong keys in our ID attack. Furthermore, we introduce the constant guessing technique in the inner rounds which significantly reduces the number of key bytes to be searched. The time and memory complexities of our attacks remain marginal.

A recent work from Eurocrypt 2023 suggests that prime-field masking has excellent potential to improve the efficiency vs. security tradeoff of masked implementations against side-channel attacks, especially in contexts where physical leakages show low noise. We pick up on the main open challenge that this seed result leads to, namely the design of an optimized prime cipher able to take advantage of this potential. Given the interest of tweakable block ciphers with cheap inverses in many leakage-resistant designs, we start by describing the FPM (Feistel for Prime Masking) family of tweakable block ciphers based on a generalized Feistel structure. We then propose a first instantiation of FPM, which we denote as small-pSquare. It builds on the recent observation that the square operation (which is non-linear in Fp) can lead to masked gadgets that are more efficient than those for multiplication, and is tailored for efficient masked implementations in hardware. We analyze the mathematical security of the FPM family of ciphers and the small-pSquare instance, trying to isolate the parts of our study that can be re-used for other instances. We additionally evaluate the implementation features of small-pSquare by comparing the efficiency vs. security tradeoff of masked FPGA circuits against those of a state-of-the art binary cipher, namely SKINNY, confirming significant gains in relevant contexts.

A Boolean function with good cryptographic properties over a set of vectors with constant Hamming weight is significant for stream ciphers like FLIP [MJSC16]. This paper presents a construction weightwise almost perfectly balanced (WAPB) Boolean functions by perturbing the support vectors of a highly nonlinear function in the construction presented in [DM]. As a result, the nonlinearity and weightwise nonlinearities of the modified functions improve substantially.

At EUROCRYPT 2017, Grassi et al. proposed the multiple-of-8 property for 5-round AES, where the number $n$ of right pairs is a multiple of 8. At ToSC 2019, Boura et al. generalized the multiple-of property for a general SPN block cipher and applied it to block cipher SKINNY. In this paper, we present that $n$ is not only a multiple but also a fixed value for SKINNY. Unlike the previous proof of generalization of multiple-of property using equivalence class, we investigate the propagation of the set to compute the exact number $n$. We experimentally verified that presented property holds. We extend this property one round more using the lack of the whitening key on the SKINNY and use this property to construct 6-round distinguisher on SKINNY-64 and SKINNY-128. The probability of success of both distinguisher is almost 1 and the total complexities are $2^{16}$ and $2^{32}$ respectively. We verified that this property only holds for SKINNY, not for AES and MIDORI, and provide the conditions under which it exists for AES-like ciphers.

An average-case variant of the $k$-SUM conjecture asserts that finding $k$ numbers that sum to 0 in a list of $r$ random numbers, each of the order $r^k$, cannot be done in much less than $r^{\lceil k/2 \rceil}$ time. On the other hand, in the dense regime of parameters, where the list contains more numbers and many solutions exist, the complexity of finding one of them can be significantly improved by Wagner's $k$-tree algorithm. Such algorithms for $k$-SUM in the dense regime have many applications, notably in cryptanalysis. In this paper, assuming the average-case $k$-SUM conjecture, we prove that known algorithms are essentially optimal for $k= 3,4,5$. For $k>5$, we prove the optimality of the $k$-tree algorithm for a limited range of parameters. We also prove similar results for $k$-XOR, where the sum is replaced with exclusive or. Our results are obtained by a self-reduction that, given an instance of $k$-SUM which has a few solutions, produces from it many instances in the dense regime. We solve each of these instances using the dense $k$-SUM oracle, and hope that a solution to a dense instance also solves the original problem. We deal with potentially malicious oracles (that repeatedly output correlated useless solutions) by an obfuscation process that adds noise to the dense instances. Using discrete Fourier analysis, we show that the obfuscation eliminates correlations among the oracle's solutions, even though its inputs are highly correlated.

A secure $n$-bit tweakable block cipher~(TBC) using $t$-bit tweaks can be modeled as a tweakable uniform random permutation, where each tweak defines an independent random $n$-bit permutation. When an input to this tweakable permutation is fixed, it can be viewed as a perfectly secure $t$-bit random function. On the other hand, when a tweak is fixed, it can be viewed as a perfectly secure $n$-bit random permutation, and it is well known that the sum of two random permutations is pseudorandom up to $2^n$ queries. A natural question is whether one can construct a pseudorandom function~(PRF) beyond the block and the tweak length bounds using a small number of calls to the underlying tweakable permutations. A straightforward way of constructing a PRF from tweakable permutations is to xor the outputs from two tweakable permutations with $c$ bits of the input to each permutation fixed. Using the multi-user security of the sum of two permutations, one can prove that the $(t+n-c)$-to-$n$ bit PRF is secure up to $2^{n+c}$ queries. In this paper, we propose a family of PRF constructions based on tweakable permutations, dubbed $\mathsf{XoTP}_c$, achieving stronger security than the straightforward construction. $\mathsf{XoTP}_c$ is parameterized by $c$, giving a $(t+n-c)$-to-$n$ bit PRF. When $t<3n$ and $c=\frac{t}{3}$, $\mathsf{XoTP}_{\frac{t}{3}}$ becomes an $(n+\frac{2t}{3})$-to-$n$ bit pseudorandom function, which is secure up to $2^{n+\frac{2t}{3}}$ queries. It provides security beyond the block and the tweak length bounds, making two calls to the underlying tweakable permutations. In order to prove the security of $\mathsf{XoTP}_c$, we extend Mirror theory to $q \gg 2^n$, where $q$ is the number of equations. From a practical point of view, our construction can be used to construct TBC-based MAC finalization functions and CTR-type encryption modes with stronger provable security compared to existing schemes.

Security proofs of symmetric-key primitives typically consider an idealized world with access to a (uniformly) random function. The starting point of our work is the observation that such an ideal world can lead to underestimating the actual security of certain primitives. As a demonstrating example, $\mathsf{XoP2}$, which relies on two independent random permutations, has been proven to exhibit superior concrete security compared to $\mathsf{XoP}$, which employs a single permutation with domain separation. But the main reason for this is an artifact of the idealized model used in the proof, in particular, that (in the random-function-ideal world) $\mathsf{XoP}$ might hit a trivially bad event (outputting $\mathbf{0}$), which does not occur in the real/domain-separated world. Motivated by this, we put forth the analysis of such primitives in an updated ideal world, which we call the {\it fine-tuned} setting, where the above artifact is eliminated. We provide fine-tuned (and enhanced) security analyses for $\mathsf{XoP}$ and $\mathsf{XoP}$-based MACs: $\mathsf{nEHtM}$ and $\mathsf{DbHtS}$, and demonstrate how the transition to the fine-tuned setting can also yield better privacy/authentication tradeoff in authenticated encryption via the generic composition of encryption and MAC. Our analyses demonstrate that the security of $\mathsf{XoP}$-based and $\mathsf{XoP2}$-based constructions are, in fact, far more similar than what was previously proven. Our security proofs rely on a fine-tuned and extended version of Mirror theory for both lower and upper bounds, which yields more versatile and improved security proofs. Of independent interest, this extension allows us to prove the multi-user MAC security of $\mathsf{nEHtM}$ in the nonce-misuse model, while the previous analysis only applied to the multi-user PRF security in the nonce-respecting model. As a side note, we also point out (and fix) a flaw in the original analysis of Chen et al. [CRYPTO'23].

In this paper, we extend the applicability of differential meet- in-the-middle attacks, proposed at Crypto 2023, to truncated differen- tials, and in addition, we introduce three new ideas to improve this type of attack: we show how to add longer structures than the original pa- per, we show how to improve the key recovery steps by introducing some probability in them, and we combine this type of attacks with the state- test technique, that was introduced in the context of impossible differ- ential attacks. Furthermore, we have developed a MILP-based tool to automate the search for a truncated differential-MITM attack with op- timized overall complexity, incorporating some of the proposed improve- ments. Thanks to this, we can build the best known attacks on the cipher CRAFT, reaching 23 rounds against 21 previously; we provide a new at- tack on 23-round SKINNY-64-192, and we improve the best attacks on SKINNY-128-384.

The tweakable Even-Mansour construction yields a tweakable block cipher from a public random permutation. We prove post-quantum security of tweakable Even-Mansour when attackers have quantum access to the random permutation but only classical access to the secretly-keyed construction, the relevant setting for most real-world applications. We then use our results to prove post-quantum security—in the same model—of the symmetric-key schemes Chaskey (an ISO-standardized MAC), Elephant (an AEAD finalist of NIST's lightweight cryptography standardization effort), and a variant of Minalpher (an AEAD second-round candidate of the CAESAR competition).

Cubic bent Boolean functions (i.e. bent functions of algebraic degree at most 3) have the property that, for every nonzero element $a$ of $\mathbb{F}_2^n$, the derivative $D_af(x)=f(x)+f(x+a)$ of $f$ admits at least one derivative $D_bD_af(x)=f(x)+f(x+a)+f(x+b)+f(x+a+b)$ that is equal to constant function 1. We study the general class of those Boolean functions having this property, which we call cubic-like bent. We study the properties of such functions and the structure of their constant second-order derivatives. We characterize them by means of their Walsh transform (that is, by their duals), by the Walsh transform of their derivatives and by other means. We study them within the Maiorana-McFarland class of bent functions, providing characterizations and constructions and showing the existence of cubic-like bent functions of any algebraic degree between 2 and $\frac n2$.

Recent advancements in post-quantum cryptography have highlighted signature schemes based on the MPC-in-the-Head (MPCitH) framework due to their reliance only on the one-way function of the underlying primitive. This reliance offers a diverse set of assumptions regarding the difficulty of post-quantum cryptographic problems. In this context, Kim et al. proposed $\mathsf{AIM}$, an MPCitH-compatible one-way function. This function is distinguished by its large algebraic S-boxes and parallel architecture, contributing to the reduced size of signatures, as presented at CCS 2023. However, $\mathsf{AIM}$ has faced several cryptanalytic challenges, which have potentially weakened its security by up to 15 bits. This paper provides a comprehensive overview of these cryptanalytic methods and proposes $\mathsf{AIM2}$, an enhanced version that addresses these identified vulnerabilities. We conduct an extensive analysis of its resilience to algebraic attacks and detail the modifications in its efficiency.

Differential cryptanalysis is an old and powerful attack against block ciphers. While different techniques have been introduced throughout the years to improve the complexity of this attack, the key recovery phase remains a tedious and error-prone procedure. In this work, we propose a new algorithm and its associated tool that permits, given a distinguisher, to output an efficient key guessing strategy. Our tool can be applied to SPN ciphers whose linear layer consists of a bit-permutation and whose key schedule is linear or almost linear. It can be used not only to help cryptanalysts find the best differential attack on a given cipher but also to assist designers in their security analysis. We applied our tool to four targets: RECTANGLE, PRESENT-80, SPEEDY-7-192 and GIFT-64. We extend the previous best attack on RECTANGLE-128 by one round and the previous best differential attack against PRESENT-80 by 2 rounds. We improve a previous key recovery step in an attack against SPEEDY and present more efficient key recovery strategies for RECTANGLE-80 and GIFT. Our tool outputs the results in only a second for most targets.

Phase-shift fault attack is a type of fault attack used for cryptanalysis of stream ciphers. It involves clocking a cipher’s feedback shift registers out of phase, in order to generate faulted keystream. Grain- 128 cipher is a 128-bit modification of the Grain cipher which is one of the finalists in the eSTREAM project. In this work, we propose a phase-shift fault attack against Grain-128 loaded with key-IV pairs that result in an all-zero LFSR after initialisation. We frame equations in terms of the input and output bits of the cipher and solve them using a SAT solver. By correctly guessing 40 internal state bits, we are able to recover the entire 128-bit key with just 2 phase-shift faults for keystreams of length 200 bits.

GCM is used in a vast amount of security protocols and is quickly becoming the de facto mode of operation for block ciphers due to its exceptional performance. In this paper we analyze the NIST stan- dardized version (SP 800-38D) of GCM, and in particular the use of short tag lengths. We show that feedback of successful or unsuccessful forgery attempt is almost always possible, contradicting the NIST assumptions for short tags. We also provide a complexity estimation of Ferguson’s authentication key recovery method on short tags, and suggest several novel improvements to Fergusons’s attacks that significantly reduce the security level for short tags. We show that for many truncated tag sizes; the security levels are far below, not only the current NIST requirement of 112-bit security, but also the old NIST requirement of 80-bit security. We therefore strongly recommend NIST to revise SP 800-38D.

Multi-user (mu) security considers large-scale attackers that, given access to a number of cryptosystem instances, attempt to compromise at least one of them. We initiate the study of mu security of the so-called GGMtree that stems from the PRG-to-PRF transformation of Goldreich, Goldwasser, and Micali, with a goal to provide references for its recently popularized use in applied cryptography. We propose a generalized model for GGM trees and analyze its mu prefix-constrained PRF security in the random oracle model. Our model allows to derive concrete bounds and improvements for various protocols, and we showcase on the Bitcoin-Improvement-Proposal standard Bip32 hierarchical wallets and function secret sharing (FSS) protocols. In both scenarios, we propose improvements with better performance and concrete security bounds at the same time. Compared with the state-of-the-art designs, our SHACAL3- and KeccaK-𝑝-based Bip32 variants reduce the communication cost of MPC-based implementations by 73.3%∼93.8%, while our AES-based FSS substantially improves mu security while reducing computations by 50%.

Authenticated Encryption (AE) modes of operation based on Tweakable Block Ciphers (TBC) usually measure efficiency in the number of calls to the underlying primitive per message block. On the one hand, many existing solutions reach a primitive-rate of 1, meaning that each n-bit block of message asymptotically needs a single call to the TBC with output length n. On the other hand, while these modes look optimal in a blackbox setting, they become less attractive when leakage comes into play, since all these calls must then be equally well protected to maintain security. Leakage-resistant modes improve this situation, by generating ephemeral keys every constant number of calls. However, rekeying is inherently suboptimal in primitive-rate, since a TBC call can only be used either to refresh a key or to encrypt a block. Even worse, existing solutions achieving almost n bits of security for n-bit secret keys have at most a primitive-rate 2/3. Hence the question: Can we design a highly-secure TBC-based rekeying mode with ``nearly optimal'' primitive-rate? We answer this question positively with Multiplex, a new mode that has primitive-rate d/(d+1) given a TBC with a dn-bit tweak. Multiplex achieves $n-\log_2(dn)$ bits of security for both (i) misuse-resilience CCA security in the blackbox setting and (ii) Ciphertext Integrity with Misuse-resistant and unbounded Leakage in encryption and decryption (CIML2). It also provides (iii) confidentiality with leakage up to the birthday bound. Furthermore, Multiplex can run d+1 calls in parallel in each iteration. The combination of these features gives a mode of operation that inherits most of the good implementation features and flexibility of a Duplex sponge -- therefore paving the way towards sound comparisons between TBC-based and permutation-based AE.

In this paper, we propose a leakage-resilient pseudo-random number generator (PRNG) design that leverages the rekeying techniques of the PSV-Enc encryption scheme and the superposition property of the Superposition-Tweak-Key (STK) framework. The random seed of the PRNG is divided into two parts; one part is used as an ephemeral key that changes every two calls to a tweakable block cipher (TBC), and the other part is used as a static long-term key. Using the superposition property, we show that it is possible to eliminate observable leakage by only masking the static key. Thus, our proposal itself can be seen as a superposition of masking and rekeying. We show that our observations can be used to design an unpredictable-with-leakage PRNG as long as the static key is protected, and the ephemeral key cannot be attacked with 2 traces. Our construction enjoys better theoretical security arguments than PSV-Enc; better Time-Data trade-off and leakage assumptions, using the recently popularized unpredictability with leakage. We verify our proposal by performing Test Vector Leakage Assessment (TVLA) on an STK-based TBC (\deoxys) operated with a fixed key and a dynamic random tweak. Our results show that while the protection of the static key is non-trivial, it only requires $\approx 10\%$ overhead for first-order protection in the most conservative setting, unlike traditional masking which may require significant overheads of $300\%$ or more.

Large language models (LLMs), exemplified by the advanced AI tool ChatGPT in 2023, have demonstrated remarkable capabilities in generating sentences, images, and program codes, driven by their development from extensive datasets. With over 100 million users worldwide, ChatGPT stands out as a leader among LLMs. Previous studies have shown its proficiency in generating program source codes for the symmetric-key block ciphers AES, CHAM, and ASCON. This study ventures into the implementation of cryptanalytic program source codes for a lightweight block cipher using ChatGPT, exploring its potential and limitations in the field of cryptography.

Hash functions are a crucial component in incrementally verifiable computation (IVC) protocols and applications. Among those, recursive SNARKs and folding schemes require hash functions to be both fast in native CPU computations and compact in algebraic descriptions (constraints). However, neither SHA-2/3 nor newer algebraic constructions, such as Poseidon, achieve both requirements. In this work we overcome this problem in several steps. First, for certain prime field domains we propose a new design strategy called Kintsugi, which explains how to construct nonlinear layers of high algebraic degree which allow fast native implementations and at the same time also an efficient circuit description for zero-knowledge applications. Then we suggest another layer, based on the Feistel Type-3 scheme, and prove wide trail bounds for its combination with an MDS matrix. We propose a new permutation design named Monolith to be used as a sponge or compression function. It is the first arithmetization-oriented function with a native performance comparable to SHA3-256. At the same time, it outperforms Poseidon in a circuit using the Merkle tree prover in the Plonky2 framework. Contrary to previously proposed designs, Monolith also allows for efficient constant-time native implementations which mitigates the risk of side-channel attacks

This note examines a nuance in the methods employed for counting the adversarial online complexity in the security proofs of duplex-based modes, with a focus on authenticated encryption. A recent study by Gilbert et al., reveals an attack on a broad class of duplex-based authenticated encryption modes. In particular, their approach to quantifying the adversarial online complexity, which capture realistic attack scenarios, includes certain queries in the count which are not in the security proofs. This note analyzes these differences and concludes that the attack of Gilbert et al, for certain parameter choices, matches the security bound.

Embedded Multimedia Cards (eMMCs) provide a protected memory area called the Replay Protected Memory Block (RPMB). eMMCs are commonly used as storage media in modern smartphones. In order to protect these devices from unauthorized access, important data is stored in the RPMB area in an authenticated manner. Modification of the RPMB data requires a pre-shared authentication key. An unauthorized user cannot change the stored data. On modern devices, this pre-shared key is generated and used exclusively within a Trusted Execution Environment (TEE) preventing attackers from access. In this paper, we investigate how the authentication key for RPMB is programmed on the eMMC. We found that this key can be extracted directly from the target memory chip. Once obtained, the authentication key can be used to manipulate stored data. In addition, poor implementation of certain security features, aimed at preventing replay attacks using RPMB on the host system can be broken by an attacker. We show how the authentication key can be extracted and how it can be used to break the anti-rollback protection to enable data restoration even after a data wipe operation has been completed. Our findings show that non-secure RPMB implementations can enable forensic investigators to break security features implemented on modern smartphones.

A cipher scheme related to ChaCha [Ber] with the variation of using 64 bit operations instead of 32 bits, and the same 512 bit state size, is presented. We will provide strong argumentation to assert that the same security of ChaCha can be obtained with less number of instructions for 24 rounds, instead of Chacha's 20 rounds. Also, an strategy to implement this cipher on SIMD extensions is presented, with a maximal throughput of about 4 bytes per cycle on a 256 bit SIMD extension with at least 11 vector registers.

DCT is a beyond-birthday-bound~(BBB) deterministic authenticated encryption~(DAE) mode proposed by Forler et al. in ACISP 2016, ensuring integrity by redundancy. The instantiation of DCT employs the BRW polynomial, which is more efficient than the usual polynomial in GCM by reducing half of the multiplication operations. However, we show that DCT suffers from a small stretch problem similar to GCM. When the stretch length $\tau$ is small, choosing a special $m$-block message, we can reduce the number of queries required by a successful forgery to $\mathcal{O}(2^{\tau}/m)$. We emphasize that this attack efficiently balances space and time complexity but does not contradict the security bounds of DCT. Finally, we propose an improved scheme named Robust DCT~(RDCT) with a minor change to DCT, which improves the security when $\tau$ is small and makes it resist the above attack.

This paper focuses on equivalences between Generalised Feistel Networks (GFN) of type-II. We introduce a new definition of equivalence which captures the concept that two GFNs are identical up to re-labelling of the inputs/outputs, and give a procedure to test this equivalence relation. Such two GFNs are therefore cryptographically equivalent for several classes of attacks. It induces a reduction of the space of possible GFNs: the set of the $(k!)^2$ possible even-odd GFNs with $2k$ branches can be partitioned into $k!$ different classes. This result can be very useful when looking for an optimal GFN regarding specific computationally intensive properties, such as the minimal number of active S-boxes in a differential trail. We also show that in several previous papers, many GFN candidates are redundant as they belong to only a few classes. Because of this reduction of candidates, we are also able to suggest better permutations than the one of WARP: they reach 64 active S-boxes in one round less and still have the same diffusion round that WARP. Finally, we also point out a new family of permutations with good diffusion properties.

In some linear key recovery attacks, the function which determines the value of the linear approximation from the plaintext, ciphertext and key is replaced by a similar map in order to improve the time or memory complexity at the cost of a data complexity increase. We propose a general framework for key recovery map substitution, and introduce Walsh spectrum puncturing, which consists of removing carefully-chosen coefficients from the Walsh spectrum of this map. The capabilities of this technique are illustrated by describing improved attacks on reduced-round Serpent (including the first 12-round attack on the 192-bit key variant), GIFT-128 and NOEKEON, as well as the full DES.

In our paper we study the effect of changing the commutative group operation used in Feistel and Lai-Massey symmetric structures into a quasigroup operation. We prove that if the quasigroup operation is isotopic with a group $\mathbb G$, the complexity of mounting a differential attack against our generalization of the Feistel structure is the same as attacking the unkeyed version of the general Feistel iteration based on $\mathbb G$. Also, when $\mathbb G$ is non-commutative we show that both versions of the Feistel structure are equivalent from a differential point of view. For the Lai-Massey structure we introduce four non-commutative versions, we argue for the necessity of working over a group and we provide some necessary conditions for the differential equivalency of the four notions.

The search for differential characteristics on block ciphers is a difficult combinatorial problem. In this paper, we investigate the performances of an AI-originated technique, Single Player Monte-Carlo Tree Search (SP-MCTS), in finding good differential characteristics on ARX ciphers, with an application to the block cipher SPECK. In order to make this approach competitive, we include several heuristics, such as the combination of forward and backward searches, and achieve significantly faster results than state-of-the-art works that are not based on automatic solvers. We reach 9, 11, 13, 13 and 15 rounds for SPECK32, SPECK48, SPECK64, SPECK96 and SPECK128 respectively. In order to build our algorithm, we revisit Lipmaa and Moriai's algorithm for listing all optimal differential transitions through modular addition, and propose a variant to enumerate all transitions with probability close (up to a fixed threshold) to the optimal, while fixing a minor bug in the original algorithm.

A systematic approach to the fixed-key analysis of differential probabilities is proposed. It is based on the propagation of 'quasidifferential trails', which keep track of probabilistic linear relations on the values satisfying a differential characteristic in a theoretically sound way. It is shown that the fixed-key probability of a differential can be expressed as the sum of the correlations of its quasidifferential trails. The theoretical foundations of the method are based on an extension of the difference-distribution table, which we call the quasidifferential transition matrix. The role of these matrices is analogous to that of correlation matrices in linear cryptanalysis. This puts the theory of differential and linear cryptanalysis on an equal footing. The practical applicability of the proposed methodology is demonstrated by analyzing several differentials for RECTANGLE, KNOT, Speck and Simon. The analysis is automated and applicable to other SPN and ARX designs. Several attacks are shown to be invalid, most others turn out to work only for some keys but can be improved for weak-keys.

PRESENT is an ultra-lightweight block cipher designed by Bogdanov et al., and has been widely studied since its proposal. It supports 80-bit and 128-bit keys, which are referred as PRESENT-80 and PRESENT-128, respectively. Up to now, linear cryptanalysis is the most effective method on attacking this cipher, especially when accelerated with the pruned Walsh transform. Combing pruned Walsh transform with multiple linear attacks, one can recover the right key for 28-round PRESENT-80 and -128. Later, this method is further improved with affine pruned Walsh transform by adding more zeros in the Walsh spectrum through rejecting some data. This leads to the 29-round attack on PRESENT-128 with full codebook. In this paper, we follow the affine pruned Walsh transform accelerated linear method, and propose 29-round attacks on both PRESENT-80 and PRESENT-128 without using full codebook. Both attacks rely on a statistical model depicting distributions of the experimental correlation when some data are artificially rejected in its computation. Besides, detailed analysis of complexity reduction for each linear hull used in attacking PRESENT is also provided and supported by an automatic tool. Our 29-round attack on PRESENT-80 mainly benefits from this tool. According to our knowledge, both attacks are the best ones on PRESENT so far.

SAT, SMT, MILP, and CP, have become prominent in the differential cryptanalysis of cryptographic primitives. In this paper, we review the techniques for constructing differential characteristic search models in these four formalisms. Additionally, we perform a systematic comparison encompassing over 20 cryptographic primitives and 16 solvers, on both easy and hard instances of optimisation, enumeration and differential probability estimation problems.

The paper presents a short survey of the History of Multivariate Cryptography together with the usage of old broken multivariate digital signatures in the new protocol based cryptosystems constructed in terms of Noncommutative Cryptography. The general schemes of New cryptosystems is a combinations of Eulerian maps and quadratic maps with their trapdoor accelerators, which are pieces of information such than the knowledge of them allow to compute the reimages in a polynomial time. These schemes are illustrated by historical examples of Imai – Matsumoto multivariate digital signatures schemes and Unbalanced Oil and Vinegar Cryptosystems.

We continue the study of $t$-wise independence of substitution-permutation networks (SPNs) initiated by the recent work of Liu, Tessaro, and Vaikuntanathan (CRYPTO 2021). Our key technical result shows that when the S-boxes are randomly and independently chosen and kept secret, an $r$-round SPN with input length $n = b \cdot k$ is $2^{-\Theta(n)}$-close to $t$-wise independent within $r = O(\min\{k, \log t\})$ rounds for any $t$ almost as large as $2^{b/2}$. Here, $b$ is the input length of the S-box and we assume that the underlying mixing achieves maximum branch number. We also analyze the special case of AES parameters (with random S-boxes), and show it is $2^{-128}$-close to pairwise independent in $7$ rounds. Central to our result is the analysis of a random walk on what we call the layout graph, a combinatorial abstraction that captures equality and inequality constraints among multiple SPN evaluations. We use our technical result to show concrete security bounds for SPNs with actual block cipher parameters and small-input $S$-boxes. (This is in contrast to the large body of results on ideal-model analyses of SPNs.) For example, for the censored-AES block cipher, namely AES with most of the mixing layers removed, we show that 192 rounds suffice to attain $2^{-128}$-closeness to pairwise independence. The prior such result for AES (Liu, Tessaro and Vaikuntanathan, CRYPTO 2021) required more than 9000 rounds.

CRAFT is a lightweight block cipher, designed to provide efficient protection against differential fault attacks. It is a tweakable cipher that includes 32 rounds to produce a ciphertext from a 64-bit plaintext using a 128-bit key and 64-bit public tweak. In this paper, compared to the designers' analysis, we provide a more detailed analysis of CRAFT against differential and zero-correlation cryptanalysis, aiming to provide better distinguishers for the reduced rounds of the cipher. Our distinguishers for reduced-round CRAFT cover a higher number of rounds compared to the designers' analysis. In our analysis, we observed that, for any number of rounds, the differential effect of CRAFT has an extremely higher probability compared to any differential trail. As an example, while the best trail for 11 rounds of the cipher has a probability of at least $2^{-80}$, we present a differential with probability $2^{-49.79}$, containing $2^{29.66}$ optimal trails, all with the same optimum probability of $2^{-80}$. Next, we use a partitioning technique, based on optimal expandable truncated trails to provide a better estimation of the differential effect on CRAFT. Thanks to this technique, we are able to find differential distinguishers for 9, 10, 11, 12, 13, and 14 rounds of the cipher in single tweak model with the probabilities of at least $2^{-40.20}$, $ 2^{-45.12} $, $ 2^{-49.79}$, $ 2^{-54.49}$, $ 2^{-59.13}$, and $ 2^{-63.80}$, respectively. These probabilities should be compared with the best distinguishers provided by the designers in the same model for 9 and 10 rounds of the cipher with the probabilities of at least $ 2^{-54.67}$ and $ 2^{-62.61}$, respectively. In addition, we consider the security of CRAFT against the new concept of related tweak zero-correlation (ZC) linear cryptanalysis and present a new distinguisher which covers 14 rounds of the cipher, while the best previous ZC distinguisher covered 13 rounds. Thanks to the related tweak ZC distinguisher for 14 rounds of the cipher, we also present 14 rounds integral distinguishers in related tweak mode of the cipher. Although the provided analysis does not compromise the cipher, we think it provides a better insight into the designing of CRAFT.

In symmetric cryptography, block ciphers, stream ciphers and permutations often make use of a round function and many round functions consist of a linear and a non-linear layer. One that is often used is based on the cellular automaton that is denoted by $\chi$ as a Boolean map on bi-infinite sequences, $\mathbb{F}^{\mathbb{Z}}$. It is defined by $\sigma \mapsto \nu$ where each $\nu_i = \sigma_i + (\sigma_{i+1}+1)\sigma_{i+2}$. A map $\chi_n$ is a map that operatos on $n$-bit arrays with periodic boundary conditions. This corresponds with $\chi$ restricted to periodic infinite sequences with period that divides $n$. This map $\chi_n$ is used in various permutations, e.g., Keccak-f (the permutation in SHA-3), ASCON (the NIST standard for lightweight cryptography), Xoodoo, Rasta and Subterranean (2.0). In this paper, we characterize the graph of $\chi$ on periodic sequences. It turns out that $\chi$ is surjective on the set of \emph{all} periodic sequences. We will show what sequences will give collisions after one application of $\chi$. We prove that, for odd $n$, the order of $\chi_n$ (in the group of bijective maps on $\mathbb{F}^n$) is $2^{\lceil \operatorname{lg}(\frac{n+1}2)\rceil}$. A given periodic sequence lies on a cycle in the graph of $\chi$, or it can be represented as a polynomial. By regarding the divisors of such a polynomial one can see whether it lies in a cycle, or after how many iterations of $\chi$ it will. Furthermore, we can see, for a given $\sigma$, the length of the cycle in its component in the state diagram. Finally, we extend the surjectivity of $\chi$ to $\mathbb{F}^{\mathbb{Z}}$, thus to include non-periodic sequences.

With the escalating demand for lightweight ciphers as well as side channel protected implementation of those ciphers in recent times, this work focuses on two related aspects. First, we present a tool for automating the task of finding a Threshold Implementation (TI) of a given Substitution Box (SBox). Our tool returns `with decomposition' and `without decomposition' based TI. The `with decomposition' based implementation returns a combinational SBox; whereas we get a sequential SBox from the `without decomposition' based implementation. Despite being high in demand, it appears that this kind of tool has been missing so far. In the process, we report new decomposition for the PRESENT SBox (improving from Poschmann et al.'s JoC'11 paper) and that of the GIFT SBox (improving from Jati et al.'s TIFS'20 paper). Second, we show an algorithmic approach where a given cipher implementation can be tweaked (without altering the cipher specification) so that its TI cost can be significantly reduced. We take the PRESENT cipher as our case study (our methodology can be applied to other ciphers as well). Indeed, we show over 31 percent reduction in area and over 52 percent reduction in depth compared to the basic threshold implementation.

Common block ciphers like AES specified by the NIST or KASUMI (A5/3) of GSM are extensively utilized by billions of individuals globally to protect their privacy and maintain confidentiality in daily communications. However, these ciphers lack comprehensive security proofs against the vast majority of known attacks. Currently, security proofs are limited to differential and linear attacks for both AES and KASUMI. For instance, the consensus on the security of AES is not based on formal mathematical proofs but on intensive cryptanalysis over its reduced rounds spanning several decades. In this work, we introduce new security proofs for AES against another attack method: impossible differential (ID) attacks. We classify ID attacks as reciprocal and nonreciprocal ID attacks. We show that sharp and generic lower bounds can be imposed on the data complexities of reciprocal ID attacks on substitution permutation networks. We prove that the minimum data required for a reciprocal ID attack on AES using a conventional ID characteristic is $2^{66}$ chosen plaintexts whereas a nonreciprocal ID attack involves at least $2^{88}$ computational steps. We mount a nonreciprocal ID attack on 6-round AES for 192-bit and 256-bit keys, which requires only $2^{18}$ chosen plaintexts and outperforms the data complexity of any attack. Given its marginal time complexity, this attack does not pose a substantial threat to the security of AES. However, we have made enhancements to the integral attack on 6-round AES, thereby surpassing the longstanding record for the most efficient attack after a period of 23 years.

This study focuses on differential cryptanalysis of the ChaCha stream cipher. In the conventional approach, an adversary first searches for an input/output differential pair with the highest differential bias and then analyzes the probabilistic neutral bits (PNB) based on the obtained input/output differential pair. However, although the time and data complexities for the attack can be estimated by the differential bias and PNB obtained by this approach, the combination of the differential bias and PNB is not always optimal. In addition, the existing studies have not performed a comprehensive analysis of the PNB; thus, they have not provided an upper bound on the number of rounds required for a differential attack that uses a single-bit truncated differential to be successful. To address these limitations, we propose a PNB-focused differential attack on reduced-round ChaCha by first comprehensively analyzing the PNB for all possible single-bit truncated output differences and then searching for the input/output differential pair with the highest differential bias based on the obtained PNB. The best existing attack on ChaCha, proposed by Beierle et al. at CRYPTO 2020, works on up to 7 rounds, whereas the most extended attack we observed works on up to 7.25 rounds using the proposed PNB-focused approach. The time complexity, data complexity, and success probability of the proposed attack are \(2^{255.62}\), \(2^{48.36}\), and 0.5, respectively. Although the proposed attack is less efficient than a brute force attack, it is the first dedicated attack on the target and provides both a baseline and useful components (i.e., differential bias and PNB) for improved attacks.

Aggregation of message authentication codes (MACs) is a proven and efficient method to preserve valuable bandwidth in resource-constrained environments: Instead of appending a long authentication tag to each message, the integrity protection of multiple messages is aggregated into a single tag. However, while such aggregation saves bandwidth, a single lost message typically means that authentication information for multiple messages cannot be verified anymore. With the significant increase of bandwidth-constrained lossy communication, as applications shift towards wireless channels, it thus becomes paramount to study the impact of packet loss on the diverse MAC aggregation schemes proposed over the past 15 years to assess when and how to aggregate message authentication. Therefore, we empirically study all relevant MAC aggregation schemes in the context of lossy channels, investigating achievable goodput improvements, the resulting verification delays, processing overhead, and resilience to denial-of-service attacks. Our analysis shows the importance of carefully choosing and configuring MAC aggregation, as selecting and correctly parameterizing the right scheme can, e.g., improve goodput by 39% to 444%, depending on the scenario. However, since no aggregation scheme performs best in all scenarios, we provide guidelines for network operators to select optimal schemes and parameterizations suiting specific network settings.

In this work we introduce algebraic transition matrices as the basis for a new approach to integral cryptanalysis that unifies monomial trails (Hu et al., Asiacrypt 2020) and parity sets (Boura and Canteaut, Crypto 2016). Algebraic transition matrices allow for the computation of the algebraic normal form of a primitive based on the algebraic normal forms of its components by means of well-understood operations from linear algebra. The theory of algebraic transition matrices leads to better insight into the relation between integral properties of $F$ and $F^{−1}$. In addition, we show that the link between invariants and eigenvectors of correlation matrices (Beyne, Asiacrypt 2018) carries over to algebraic transition matrices. Finally, algebraic transition matrices suggest a generalized definition of integral properties that subsumes previous notions such as extended division properties (Lambin, Derbez and Fouque, DCC 2020). On the practical side, a new algorithm is described to search for these generalized properties and applied to Present, resulting in new properties. The algorithm can be instantiated with any existing automated search method for integral cryptanalysis.

Filter permutators are a family of stream cipher designs that are aimed for hybrid homomorphic encryption. While originally operating on bits, they have been generalized to groups at Asiacrypt 2022, and instantiated for evaluation with the TFHE scheme which favors a filter based on (negacyclic) Look Up Tables (LUTs). A recent work of Gilbert et al., to appear at Asiacrypt 2023, exhibited (algebraic) weaknesses in the Elisabeth-4 instance, exploiting the combination of the 4-bit negacyclic LUTs it uses as filter. In this article, we explore the landscape of patches that can be used to restore the security of such designs while maintaining their good properties for hybrid homomorphic encryption. Starting with minimum changes, we observe that just updating the filter function (still with small negacyclic LUTs) is conceptually feasible, and propose the resulting Elisabeth-b4 design with three levels of NLUTs. We then show that a group permutator combining two different functions in the filter can simplify the analysis and improve performances. We specify the Gabriel instance to illustrate this claim. We finally propose to modify the group filter permutator paradigm into a mixed filter permutator, which considers the permutation of the key with elements in a group and a filter outputting elements in a different group. We specify the Margrethe instance as a first example of mixed filter permutator, with key elements in $\mathbb{F}_2$ and output in $\mathbb{Z}_{16}$, that we believe well-suited for recent fully homomorphic encryption schemes that can efficiently evaluate larger (not negacyclic) LUTs.

QCB is a proposal for a post-quantum secure, rate-one authenticated encryption with associated data scheme (AEAD) based on classical OCB3 and \(\Theta\)CB, which are vulnerable against a quantum adversary in the Q2 setting. The authors of QCB prove integrity under plus-one unforgeability, whereas the proof of the stronger definition of blind unforgeability has been left as an open problem. After a short overview of QCB and the current state of security definitions for authentication, this work proves blind unforgeability of QCB. Finally, the strategy of using tweakable block ciphers in authenticated encryption is generalised to a generic blindly unforgeable AEAD model.

In 2019, Essaid et al. introduced a chaotic map-based encryption scheme for color images. Their approach employs three improved chaotic maps to dynamically generate the key bytes and matrix required by the cryptosystem. It should be noted that these parameters are dependent on the size of the source image. According to the authors, their method offers adequate security (i.e. $279$ bits) for transmitting color images over unsecured channels. However, we show in this paper that this is not the case. Specifically, we present two cryptanalytic attacks that undermine the security of Essaid et al.'s encryption scheme. In the case of the chosen plaintext attack, we require only two chosen plaintexts to completely break the scheme. The second attack is a a chosen ciphertext attack, which requires two chosen ciphertexts and compared to the first one has a rough complexity of $2^{24}$. The attacks are feasible due to the fact that the key bits and matrix generated by the algorithm remain unaltered for distinct plaintext images.

This paper presents a provably secure, higher-order, and leakage-resilient (LR) rekeying scheme named LR Rekeying with Random oracle Repetition (LR4), along with a quantitative security evaluation methodology. Many existing LR primitives are based on a concept of leveled implementation, which still essentially require a leak-free sanctuary (i.e., differential power analysis (DPA)-resistant component(s)) for some parts. In addition, although several LR pseudorandom functions (PRFs) based on only bounded DPA-resistant components have been developed, their validity and effectiveness for rekeying usage still need to be determined. In contrast, LR4 is formally proven under a leakage model that captures the practical goal of side-channel attack (SCA) protection (e.g., masking with a practical order) and assumes no unbounded DPA-resistant sanctuary. This proof suggests that LR4 resists exponential invocations (up to the birthday bound of key size) without using any unbounded leak-free component, which is the first of its kind. Moreover, we present a quantitative SCA success rate evaluation methodology for LR4 that combines the bounded leakage models for LR cryptography and a state-of-the-art information-theoretical SCA evaluation method. We validate its soundness and effectiveness as a DPA countermeasure through a numerical evaluation; that is, the number of secure calls of a symmetric primitive increases exponentially by increasing a security parameter under practical conditions.

Zero-knowledge (ZK) applications form a large group of use cases in modern cryptography, and recently gained in popularity due to novel proof systems. For many of these applications, cryptographic hash functions are used as the main building blocks, and they often dominate the overall performance and cost of these approaches. Therefore, in the last years several new hash functions were built in order to reduce the cost in these scenarios, including Poseidon and Rescue among others. These hash functions often look very different from more classical designs such as AES or SHA-2. For example, they work natively over prime fields rather than binary ones. At the same time, for example Poseidon and Rescue share some common features, such as being SPN schemes and instantiating the nonlinear layer with invertible power maps. While this allows the designers to provide simple and strong arguments for establishing their security, it also introduces crucial limitations in the design, which may affect the performance in the target applications. In this paper, we propose the Horst construction, in which the addition in a Feistel scheme (x, y) -> (y + F(x), x) is extended via a multiplication, i.e., (x, y) -> (y * G(x) + F(x), x). By carefully analyzing the performance metrics in SNARK and STARK protocols, we show how to combine an expanding Horst scheme with a Rescue-like SPN scheme in order to provide security and better efficiency in the target applications. We provide an extensive security analysis for our new design Griffin and a comparison with all current competitors.

This note presents attacks on the lightweight hash function TS-Hash proposed by Tsaban, including a polynomial-time preimage attack for short messages (at most n/2 bits), high-probability differentials, a general subexponential-time preimage attack, and linearization techniques.

The LowMC family of SPN block cipher proposed by Albrecht et al. was designed specifically for MPC-/FHE-/ZKP-friendly use cases. It is especially used as the underlying block cipher of PICNIC, one of the alternate third-round candidate digital signature algorithms for NIST post-quantum cryptography standardization. The security of PICNIC is highly related to the difficulty of recovering the secret key of LowMC from a given plaintext/ciphertext pair, which raises new challenges for security evaluation under extremely low data complexity. In this paper, we improve the attacks on LowMC under low data complexity, i.e. 1 or 2 chosen plaintext/ciphertext pairs. For the difference enumeration attack with 2 chosen plaintexts, we propose new algebraic methods to better exploit the nonlinear relation inside the introduced variables based on the attack framework proposed by Liu et al. at ASIACRYPT 2022. With this technique, we significantly extend the number of attack rounds for LowMC with partial nonlinear layers and improve the success probability from around 0.5 to over 0.9. The security margin of some instances can be reduced to only 3/4 rounds. For the key-recovery attack using a single plaintext, we adopt a different linearization strategy to reduce the huge memory consumption caused by the polynomial methods for solving multivariate equation systems. The memory complexity reduces drastically for all 5-/6-round LowMC instances with full nonlinear layers at the sacrifice of a small factor of time complexity. For 5-round LowMC instances with a block size of 129, the memory complexity decreases from $2^{86.46}$ bits to $2^{48.18}$ bits while the time complexity even slightly reduces. Our results indicate that the security for different instances of LowMC under extremely low data complexity still needs further exploration.

The linear layer of block ciphers plays an important role in their security. In particular, ciphers designed following the wide-trail strategy use the branch number of the linear layer to derive bounds on the probability of linear and differential trails. At FSE 2014, the LS-design construction was introduced as a simple and regular structure to design bitsliced block ciphers. It considers the internal state as a bit matrix, and applies alternatively an identical S-Box on all the columns, and an identical L-Box on all the lines. Security bounds are derived from the branch number of the L-Box. In this paper, we focus on bitsliced linear layers inspired by the LS-design construction and the Spook AEAD algorithm. We study the construction of bitsliced linear transformations with efficient implementations using XORs and rotations (optimized for bitsliced ciphers implemented on 32-bit processors), and a high branch number. In order to increase the density of the activity patterns, the linear layer is designed on the whole state, rather than using multiple parallel copies of an L-Box. Our main result is a linear layer for 128-bit ciphers with branch number 21, improving upon the best 32-bit transformation with branch number 12, and the one of Spook with branch number 16.

A backdoored Pseudorandom Generator (PRG) is a PRG which looks pseudorandom to the outside world, but a saboteur can break PRG security by planting a backdoor into a seemingly honest choice of public parameters, $pk$, for the system. Backdoored PRGs became increasingly important due to revelations about NIST’s backdoored Dual EC PRG, and later results about its practical exploitability. Motivated by this, at Eurocrypt'15 Dodis et al. [21] initiated the question of immunizing backdoored PRGs. A $k$-immunization scheme repeatedly applies a post-processing function to the output of $k$ backdoored PRGs, to render any (unknown) backdoors provably useless. For $k=1$, [21] showed that no deterministic immunization is possible, but then constructed "seeded" $1$-immunizer either in the random oracle model, or under strong non-falsifiable assumptions. As our first result, we show that no seeded $1$-immunization scheme can be black-box reduced to any efficiently falsifiable assumption. This motivates studying $k$-immunizers for $k\ge 2$, which have an additional advantage of being deterministic (i.e., "seedless"). Indeed, prior work at CCS'17 [37] and CRYPTO'18 [7] gave supporting evidence that simple $k$-immunizers might exist, albeit in slightly different settings. Unfortunately, we show that simple standard model proposals of [37, 7] (including the XOR function [7]) provably do not work in our setting. On a positive, we confirm the intuition of [37] that a (seedless) random oracle is a provably secure $2$-immunizer. On a negative, no (seedless) $2$-immunization scheme can be black-box reduced to any efficiently falsifiable assumption, at least for a large class of natural $2$-immunizers which includes all "cryptographic hash functions." In summary, our results show that $k$-immunizers occupy a peculiar place in the cryptographic world. While they likely exist, and can be made practical and efficient, it is unlikely one can reduce their security to a "clean" standard-model assumption.

A Key Derivation Function (KDF) generates a uniform and highly random key-stream from weakly random key material. KDFs are broadly used in various security protocols such as digital signatures and key exchange protocols. HKDF, the most deployed KDF in practice, is based on the extract-then-expand paradigm. It is presently used, among others, in the Signal Protocol for end-to-end encrypted messaging. HKDF is a generic KDF for general input sources and thus is not optimized for source-specific use cases such as key derivation from Diffie-Hellman (DH) sources (i.e. DH shared secrets as key material). Furthermore, the sequential HKDF design is unnecessarily slow on some general-purpose platforms that can benefit from parallelization. In this work, we propose a novel, efficient and secure KDF called $\mathsf{Skye}$. $\mathsf{Skye}$ follows the extract-then-expand paradigm and consists of two algorithms: efficient deterministic randomness extractor and expansion functions. Instantiating our extractor for dedicated source-specific (e.g. DH sources) inputs leads to a significant efficiency gain over HKDF while maintaining the security level. We provide concrete security analysis of $\mathsf{Skye}$ and both its algorithms in the standard model. We provide a software performance comparison of $\mathsf{Skye}$ with the AES-based expanding PRF $\mathsf{ButterKnife}$ and HKDF with SHA-256 (as used in practice). Our results show that in isolation $\mathsf{Skye}$ performs from $4\text{x}$ to $47\text{x}$ faster than HKDF, depending on the availability of AES or SHA instruction support. We further demonstrate that with such a performance gain, when $\mathsf{Skye}$ is integrated within the current Signal implementation, we can achieve significant overall improvements ranging from $38\%$ to $64\%$ relative speedup in unidirectional messaging. Even in bidirectional messaging, that includes DH computation with dominating computational cost, $\mathsf{Skye}$ still contributes to $12$-$36\%$ relative speedup when just $10$ messages are sent and received at once.

On modern processors HCTR is one of the most efficient constructions for building a tweakable super-pseudorandom permutation. However, a bug in the specification and another in Chakraborty and Nandi's security proof invalidate the claimed security bound. We here present HCTR2, which fixes these issues and improves the security bound, performance and flexibility. GitHub: https://github.com/google/hctr2

In this paper, we analyze the Espresso cipher from a related key chosen IV perspective. More precisely, we explain how one can obtain Key-IV pairs such that Espresso's keystreams either have certain identical bits or are shifted versions of each other. For the first case, we show how to obtain such pairs after $2^{32}$ iterations, while for the second case, we present an algorithm that produces such pairs in $2^{28}$ iterations. Moreover, we show that by making a minor change in the padding used during the initialization phase, it can lead to a more secure version of the cipher. Specifically, changing the padding increases the complexity of our second attack from $2^{28}$ to $2^{34}$. Finally, we show how related IVs can accelerate brute force attacks, resulting in a faster key recovery. Although our work does not have any immediate implications for breaking the Espresso cipher, these observations are relevant in the related-key chosen IV scenario.

In this paper, inspired by the work of Beyne and Rijmen at CRYPTO 2022, we explore the accurate probability of $d$-differential in the fixed-key model. The theoretical foundations of our method are based on a special matrix $-$ quasi-$d$-differential transition matrix, which is a natural extension of the quasidifferential transition matrix. The role of quasi-$d$-differential transition matrices in polytopic cryptananlysis is analogous to that of correlation matrices in linear cryptanalysis. Therefore, the fixed-key probability of a $d$-differential can be exactly expressed as the sum of the correlations of its quasi-$d$-differential trails. Then we revisit the boomerang attack from a perspective of 3-differential. Different from previous works, the probability of a boomerang distinguisher can be exactly expressed as the sum of the correlations of its quasi-$3$-differential trails without any assumptions in our work. In order to illustrate our theory, we apply it to the lightweight block cipher GIFT. It is interesting to find the probability of every optimal 3-differential characteristic of an existing 2-round boomerang is zero, which can be seen as an evidence that the security of block ciphers adopting half-round key XOR might be overestimated previously to some extent in differential-like attacks.

Weightwise degree-d functions are Boolean functions that take the values of a function of degree at most d on each set of fixed Hamming weight. The class of weightwise affine functions encompasses both the symmetric functions and the Hidden Weight Bit Function (HWBF). The good cryptographic properties of the HWBF, except for the nonlinearity, motivates to investigate a larger class with functions that share the good properties and have a better nonlinearity. Additionally, the homomorphic friendliness of symmetric functions exhibited in the context of hybrid homomorphic encryption and the recent results on homomorphic evaluation of Boolean functions make this class of functions appealing for efficient privacy-preserving protocols. In this article we realize the first study on weightwise degree-d functions, focusing on weightwise affine and weightwise quadratic functions. We show some properties on these new classes of functions, in particular on the subclass of cyclic weightwise functions. We provide balanced constructions and prove nonlinearity lower bounds for all cyclic weightwise affine functions and for a family of weightwise quadratic functions. We complement our work with experimental results, they show that other cyclic weightwise linear functions than the HWBF have better cryptographic parameters, and considering weightwise quadratic functions allows to reach higher algebraic immunity and substantially better nonlinearity.

This paper makes a comprehensive study of two important strategies for polynomial hashing over a prime order field $\mathbb{F}_p$, namely usual polynomial based hashing and hashing based on Bernstein-Rabin-Winograd (BRW) polynomials, and the various ways to combine them. Several hash functions are proposed and upper bounds on their differential probabilities are derived. Concrete instantiations are provided for the primes $p=2^{127}-1$ and $p=2^{130}-5$. A major contribution of the paper is an extensive 64-bit implementation of all the proposed hash functions in assembly targeted at modern Intel processors. The timing results suggest that using the prime $2^{127}-1$ is significantly faster than using the prime $2^{130}-5$. Further, a judicious mix of the usual polynomial based hashing and BRW-polynomial based hashing can provide a significantly faster alternative to only usual polynomial based hashing. In particular, the timing results of our implementations show that our final hash function proposal for the prime $2^{127}-1$ is much faster than the well known Poly1305 hash function defined over the prime $2^{130}-5$, achieving speed improvements of up to 40%.

Sponge based constructions have gained significant popularity for designing lightweight authenticated encryption modes. Most of the authenticated ciphers following the Sponge paradigm can be viewed as variations of the Transform-then-permute construction. It is known that a construction following the Transform-then-permute paradigm provides security against any adversary having data complexity $D$ and time complexity $T$ as long as $DT \ll 2^{b-r}$. Here, $b$ represents the size of the underlying permutation, while $r$ pertains to the rate at which the message is injected. The above result demonstrates that an increase in the rate leads to a degradation in the security of the constructions, with no security guaranteed to constructions operating at the full rate, where $r=b$. This present study delves into the exploration of whether adding some auxiliary states could potentially improve the security of the Transform-then-permute construction. Our investigation yields an affirmative response, demonstrating that a special class of full rate Transform-then-permute with additional states, dubbed frTtP+, can indeed attain security when operated under a suitable feedback function and properly initialized additional state. To be precise, we prove that frTtP+ provides security as long as $D \ll 2^{s/2}$ and $T \ll 2^{s}$, where $s$ denotes the size of the auxiliary state in terms of bits. To demonstrate the applicability of this result, we show that the construction $Orange-Zest_{mod}$ belongs to this class, thereby obtaining the desired security. In addition, we propose a family of full-rate Transform-then-permute construction with a Beetle-like feedback function, dubbed \textsf{fr-Beetle}, which also achieves the same level of security.

We design a SNARKs/STARKs-optimized AEAD scheme based on the $\texttt{MonkeySpongeWrap}$ (ToSC 2023(2)) and the RPO permutation (ePrint 2022/1577).

The Ascon cipher suite, offering both authenticated encryption with associated data (AEAD) and hashing functionality, has recently emerged as the winner of the NIST Lightweight Cryptography (LwC) standardization process. The AEAD schemes within Ascon, namely Ascon-128 and Ascon-128a, have also been previously selected as the preferred lightweight authenticated encryption solutions in the CAESAR competition. In this paper, we present a tight and comprehensive security analysis of the Ascon AEAD schemes within the random permutation model. Existing integrity analyses of Ascon (and any Duplex AEAD scheme in general) commonly include the term $DT/2^c$, where $D$ and $T$ represent data and time complexities respectively, and $c$ denotes the capacity of the underlying sponge. In this paper, we demonstrate that Ascon achieves AE security when $T$ is bounded by $\min\{2^{\kappa}, 2^c\}$ (where $\kappa$ is the key size), and $DT$ is limited to $2^b$ (with $b$ being the size of the underlying permutation, which is 320 for Ascon). Our findings indicate that in accordance with NIST requirements, Ascon allows for a tag size as low as 64 bits while enabling a higher rate of 184 bits, surpassing the recommended rate.

In TCHES’22, Shen et al. proposed Triplex, a single-pass leakage-resistant authenticated encryption scheme based on Tweakable Block Ciphers (TBCs) with 2n-bit tweaks. Triplex enjoys beyond-birthday-bound ciphertext integrity in the CIML2 setting and birthday-bound confidentiality in the CCAmL1 notion. Despite its strengths, Triplex’s operational efficiency was hindered by its sequential nature, coupled with a rate limit of 2/3. In an endeavor to surmount these efficiency challenges, Peters et al. proposed Multiplex, a variant of Triplex with increased parallelism and a flexible rate of d/(d+1) that retains similar security guarantees. However, the innovation came at the price of requiring TBCs with dn-bit tweaks, which are unusual and potentially costly for d > 3. In this paper, we investigate the limits of generalized Triplex- and Multiplex-type constructions for single-pass leakage-resilient authenticated encryption. Our contributions are threefold. First, we show that such constructions cannot provide CIML2 integrity for any tweak lengths below dn/2 bits. Second, we provide a birthday-bound attack for constructions with TBCs of tweak lengths between dn/2 and (d − 1)n + n/2 bits. Finally, on the constructive side, we propose a family of single-pass leakage-resilient authenticated ciphers, dubbed Tweplex, that uses tweaks of dn/2 bits and provides a rate of d/(d + 1) while providing n/2-bit CIML2 integrity and CCAmL1 confidentiality.

Fault attacks impose a serious threat against the practical implementations of cryptographic algorithms. Statistical Ineffective Fault Attacks (SIFA), exploiting the dependency between the secret data and the fault propagation overcame many of the known countermeasures. Later, several countermeasures have been proposed to tackle this attack using error detection methods. However, the efficiency of the countermeasures, in part governed by the number of error checks, still remains a challenge. In this work, we propose a fault countermeasure, StaTI, based on threshold implementations and linear encoding techniques. The proposed countermeasure protects the implementations of cryptographic algorithms against both side-channel and fault adversaries in a non-combined attack setting. We present a new composable notion, stability, to protect a threshold implementation against a formal gate/register-faulting adversary. Stability ensures fault propagation, making a single error check of the output suffice. To illustrate the stability notion, first, we provide stable encodings of the XOR and AND gates. Then, we present techniques to encode threshold implementations of S-boxes, and provide stable encodings of some quadratic S-boxes together with their security and performance evaluation. Additionally, we propose general encoding techniques to transform a threshold implementation of any function (e.g., non-injective functions) to a stable one. We then provide an encoding technique to use in symmetric primitives which encodes state elements together significantly reducing the encoded state size. Finally, we used StaTI to implement a secure Keccak on FPGA and report on its efficiency.

Impossible differential (ID) cryptanalysis is one of the most important attacks on block ciphers. The Mixed Integer Linear Programming (MILP) model is a popular method to determine whether a specific difference pair is an ID. Unfortunately, due to the huge search space (approximately $2^{2n}$ for a cipher with a block size $n$ bits), we cannot leverage this technique to exhaust all difference pairs, which is a well-known long-standing problem. In this paper, we propose a systematic method to find all IDs for SPN block ciphers. The idea is to partition the whole difference pair space into lots of small disjoint sets, each of which has a representative difference pair. All difference pairs in one small set are possible if its representative pair is possible, and this can be conveniently checked by the MILP model. In this way, the overall search space is drastically reduced to a practical size by excluding the sets containing no IDs. We then examine the remaining difference pairs to identify all IDs (if some IDs exist). If our method cannot find any ID, the target cipher is proved free of ID distinguishers. Our method works especially well for SPN ciphers with block size 64. We apply our method to SKINNY-64 and successfully find all 432 and 12 truncated IDs (we find all IDs but all of them can be assembled into certain truncated IDs) for 11 and 12 rounds, respectively. We also prove, for the first time, that 13-round SKINNY-64 is free of ID distinguishers even when considering the differential transitions through the Difference Distribution Table (DDT). Similarly, we find all 12 truncated IDs (all IDs are assembled into 12 truncated IDs) for 13-round CRAFT and prove there is no ID for 14 rounds. For SbPN cipher GIFT-64, we prove that there is no ID for 8 rounds. For SPN ciphers with larger block sizes, we show that our idea is also useful to strengthen the current search methods. For example, if we consider the Sbox to be ideal and only consider the branch number information of the diffusion matrix, we can find all 6,750 truncated IDs for 6-round Rijndael-192 in 1 second and prove that there is no truncated ID for 7 rounds. Previously, we need to solve approximately $2^{48}$ MILP models to achieve the same goal. For GIFT-128, we exhausted all difference patterns that have an active superbox in the plaintext and ciphertext and proved there is no ID of such patterns for 8 rounds. Although we have searched for a larger or even full space for IDs, no longer ID distinguishers have been found. This implies the reasonableness of the intuition that a small number (usually one or two) of active bits/words at the beginning and end of an ID will be the longest.

In this article we study the Algebraic Immunity (AI) of Weightwise Perfectly Balanced (WPB) functions. After showing a lower bound on the AI of two classes of WPB functions from the previous literature, we prove that the minimal AI of a WPB $n$-variables function is constant, equal to $2$ for $n\ge 4$ . Then, we compute the distribution of the AI of WPB function in $4$ variables, and estimate the one in $8$ and $16$ variables. For these values of $n$ we observe that a large majority of WPB functions have optimal AI, and that we could not obtain an AI-$2$ WPB function by sampling at random. Finally, we address the problem of constructing WPB functions with bounded algebraic immunity, exploiting a construction from 2022 by Gini and Méaux. In particular, we present a method to generate multiple WPB functions with minimal AI, and we prove that the WPB functions with high nonlinearity exhibited by Gini and Méaux also have minimal AI. We conclude with a construction giving WPB functions with lower bounded AI, and give as example a family with all elements with AI at least $n/2-\log(n)+1$.

In this article, we propose several generic methods for constructing minimal linear codes over the field $\mathbb{F}_p$. The first construction uses the method of direct sum of an arbitrary function $f:\mathbb{F}_{p^r}\to \mathbb{F}_{p}$ and a bent function $g:\mathbb{F}_{p^s}\to \mathbb{F}_p$ to induce minimal codes with parameters $[p^{r+s}-1,r+s+1]$ and minimum distance larger than $p^r(p-1)(p^{s-1}-p^{s/2-1})$. For the first time, we provide a general construction of linear codes from a subclass of non-weakly regular plateaued functions, which partially answers an open problem posed in [22]. The second construction deals with a bent function $g:\mathbb{F}_{p^m}\to \mathbb{F}_p$ and a subspace of suitable derivatives $U$ of $g$, i.e., functions of the form $g(y+a)-g(y)$ for some $a\in \mathbb{F}_{p^m}^*$. We also provide a sound generalization of the recently introduced concept of non-covering permutations [45]. Some important structural properties of this class of permutations are derived in this context. The most remarkable observation is that the class of non-covering permutations contains the class of APN power permutations (characterized by having two-to-one derivatives). Finally, the last general construction combines the previous two methods (direct sum, non-covering permutations and subspaces of derivatives), using a bent function in the Maiorana-McFarland class to construct minimal codes (even those violating the Ashikhmin-Barg bound) with larger dimensions. This last method proves to be highly flexible since it can lead to several non-equivalent codes, depending to a great extent on the choice of the underlying non-covering permutation.

The key step of the cube attack is to recover the special polynomial, the superpoly, of the target cipher. In particular, the balanced superpoly, in which there exists at least one secret variable as a single monomial and none of the other monomials contain this variable, can be exploited to reveal one-bit information about the key bits. However, as the number of rounds grows, it becomes increasingly difficult to find such balanced superpolies. Consequently, traditional methods of searching for balanced superpolies soon hit a bottleneck. Aiming at performing a cube attack on more rounds of Trivium with a practical complexity, in this paper, we present three techniques to obtain sufficient balanced polynomials. 1. Based on the structure of Trivium, we propose a variable substitution technique to simplify the superpoly. 2. Obtaining the additional balanced polynomial by combining two superpolies to cancel the two-degree terms. 3. We propose an experimental approach to construct high-quality large cubes which may contain more subcubes with balanced superpolies and a heuristic search strategy for their subcubes whose superpolies are balanced. To illustrate the power of our techniques, we search for balanced polynomials for 810- and 825-round Trivium. As a result, we can mount cube attacks against 810- and 825-round Trivium with the time complexity of $2^{44.17}$ and $2^{53.17}$ round-reduced Trivium initializations, respectively, which can be verified in 48 minutes and 18 days on a PC with one A100 GPU. For the same level of time complexity, this improves the previous best results by $2$ and $5$ rounds, respectively.

At CRYPTO'18, Datta et al. proposed nPolyMAC and proved the security up to 2^{2n/3} authentication queries and 2^{n} verification queries. At EUROCRYPT'19, Dutta et al. proposed CWC+ and showed the security up to 2^{2n/3} queries. At FSE'19, Datta et al. proposed PolyMAC and its key-reduced variant 2k-PolyMAC, and showed the security up to 2^{2n/3} queries. This security bound was then improved by Kim et al. (EUROCRYPT'20) and Datta et al (FSE'23) respectively to 2^{3n/4} and in the multi-user setting. At FSE'20, Chakraborti et al. proposed PDM*MAC and 1k-PDM*MAC and showed the security up to 2^{2n/3} queries. Recently, Chen et al. proposed nEHtM_p^+ and showed the security up to 2^{2n/3} queries. In this paper, we show forgery attacks on nPolyMAC, CWC+, PolyMAC, 2k-PolyMAC, PDM*MAC, 1k-PDM*MAC and nEHtM_p^+. Our attacks exploit some vulnerability in the underlying polynomial hash function Poly, and (i) require only one authentication query and one verification query; (ii) are nonce-respecting; (iii) succeed with probability 1. Thus, our attacks disprove the provable high security claims of these schemes. We then revisit their security analyses and identify what went wrong. Finally, we propose two solutions that can restore the beyond-birthday-bound security.

In ASIACRYPT'17, Naito proposed a beyond-birthday-bound variant of the LightMAC construction, called LightMAC_Plus, which is built on three independently keyed $n$-bit block ciphers, and showed that the construction achieves $2n/3$-bits PRF security. Later, Kim et al. claimed (without giving any formal proof) its security bound to $2^{3n/4}$. In FSE'18, Datta et al. have proposed a two-keyed variant of the LightMAC_Plus construction, called 2k-LightMAC_Plus, which is built on two independently keyed $n$-bit block ciphers, and showed that the construction achieves $2n/3$-bits PRF security. In this paper, we show a tight security bound on the 2k-LightMAC_Plus construction. In particular, we show that it provably achieves security up to $2^{3n/4}$ queries. We also exhibit a matching attack on the construction with the same query complexity and hence establishing the tightness of the security bound. To the best of our knowledge, this is the first work that provably shows a message length independent $3n/4$-bit tight security bound on a block cipher based variable input length PRF with two block cipher keys.

The National Cyber Security Centre (NCSC) is the government organisation responsible for mitigating cyber security risks to the UK. Our work securing UK public- and private-sector networks involves (amongst many other security measures) research into cryptographic design, primarily to protect data requiring long-term security or data for which we have a particularly low tolerance of risk to its transmission and storage. Our algorithms prioritise robustness over other important considerations, such as performance, more highly than other designs. We present GLEVIAN and VIGORNIAN: two AEAD modes with proofs of beyond-birthday security, security against nonce misuse, and against the release of unverified plaintext – both of the latter in strong notions of these security properties. We discuss our hierarchy of requirements for AEAD modes, and the rationale for the design choices made. GLEVIAN and VIGORNIAN demonstrate we can achieve significantly improved robustness over GCM for use cases where some performance degradation is acceptable. We are not aware of other designs offering exactly the security properties of GLEVIAN and VIGORNIAN, and are publishing our designs to support the research that will inform the recently announced effort by NIST to standardise new modes of operation. We believe our work could be of interest to those with use cases similar to ours, and we offer suggestions for future research that might build on the work in this paper.

The Higher-order Differential-Linear (HDL) attack was introduced by Biham \textit{et al.} at FSE 2005, where a linear approximation was appended to a Higher-order Differential (HD) transition. It is a natural generalization of the Differential-Linear (DL) attack. Due to some practical restrictions, however, HDL cryptanalysis has unfortunately attracted much less attention compared to its DL counterpart since its proposal. In this paper, we revisit HD/HDL cryptanalysis from an algebraic perspective and provide two novel tools for detecting possible HD/HDL distinguishers, including: (a) Higher-order Algebraic Transitional Form (HATF) for probabilistic HD/HDL attacks; (b) Differential Supporting Function (\DSF) for deterministic HD attacks. In general, the HATF can estimate the biases of $\ell^{th}$-order HDL approximations with complexity $\mathcal{O}(2^{\ell+d2^\ell})$ where $d$ is the algebraic degree of the function studied. If the function is quadratic, the complexity can be further reduced to $\mathcal{O}(2^{3.8\ell})$. HATF is therefore very useful in HDL cryptanalysis for ciphers with quadratic round functions, such as \ascon and \xoodyak. \DSF provides a convenient way to find good linearizations on the input of a permutation, which facilitates the search for HD distinguishers. Unsurprisingly, HD/HDL attacks have the potential to be more effective than their simpler differential/DL counterparts. Using HATF, we found many HDL approximations for round-reduced \ascon and \xoodyak initializations, with significantly larger biases than DL ones. For instance, there are deterministic 2$^{nd}$-order/4$^{th}$-order HDL approximations for \ascon/\xoodyak initializations, respectively (which is believed to be impossible in the simple DL case). We derived highly biased HDL approximations for 5-round \ascon up to 8$^{th}$ order, which improves the complexity of the distinguishing attack on 5-round \ascon from $2^{16}$ to $2^{12}$ calls. We also proposed HDL approximations for 6-round \ascon and 5-round \xoodyak (under the single-key model), which couldn't be reached with simple DL so far. For key recovery, HDL attacks are also more efficient than DL attacks, thanks to the larger biases of HDL approximations. Additionally, HATF works well for DL (1$^{st}$-order HDL) attacks and some well-known DL biases of \ascon and \xoodyak that could only be obtained experimentally before can now be predicted theoretically. With \DSF, we propose a new distinguishing attack on 8-round \ascon permutation, with a complexity of $2^{48}$. Also, we provide a new zero-sum distinguisher for the full 12-round \ascon permutation with $2^{55}$ time/data complexity. We highlight that our cryptanalyses do not threaten the security of \ascon or \xoodyak.

This paper shows how to achieve a quantum speed-up for multidimensional (zero correlation) linear distinguishers. A previous work by Kaplan et al. has already shown a quantum quadratic speed-up for one-dimensional linear distinguishers. However, classical linear cryptanalysis often exploits multidimensional approximations to achieve more efficient attacks, and in fact it is highly non-trivial whether Kaplan et al.'s technique can be extended into the multidimensional case. To remedy this, we investigate a new quantum technique to speed-up multidimensional linear distinguishers. Firstly, we observe that there is a close relationship between the subroutine of Simon's algorithm and linear correlations via Fourier transform. Specifically, a slightly modified version of Simon's subroutine, which we call Correlation Extraction Algorithm (CEA), can be used to speed-up multidimensional linear distinguishers. CEA also leads to a speed-up for multidimensional zero correlation distinguishers, as well as some integral distinguishers through the correspondence of zero correlation and integral properties shown by Bogdanov et al.~and Sun et al. Furthermore, we observe possibility of a more than quadratic speed-ups for some special types of integral distinguishers when multiple integral properties exist. Especially, we show a single-query distinguisher on a 4-bit cell SPN cipher with the same integral property as 2.5-round AES. Our attacks are the first to observe such a speed-up for classical cryptanalytic techniques without relying on hidden periods or shifts. By replacing the Hadamard transform in CEA with the general quantum Fourier transform, our technique also speeds-up generalized linear distinguishers on an arbitrary finite abelian group.

We provide security foundations for SAFE, a recently introduced API framework for sponge-based hash functions tailored to prime-field-based protocols. SAFE aims to provide a robust and foolproof interface, has been implemented in the Neptune hash framework and some zero-knowledge proof projects, but currently lacks any security proof. In this work we identify the SAFECore as versatile variant sponge construction underlying SAFE, we prove indifferentiability of SAFECore for all (binary and prime) fields up to around $|\mathbb{F}_p|^{c/2}$ queries, where $\mathbb{F}_p$ is the underlying field and $c$ the capacity, and we apply this security result to various use cases. We show that the SAFE-based protocols of plain hashing, authenticated encryption, verifiable computation, non-interactive proofs, and commitment schemes are secure against a wide class of adversaries, including those dealing with multiple invocations of a sponge in a single application. Our results pave the way of using SAFE with the full taxonomy of hash functions, including SNARK-, lattice-, and x86-friendly hashes.

We introduce the QARMAv2 family of tweakable block ciphers. It is a redesign of QARMA (from FSE 2017) to improve its security bounds and allow for longer tweaks, while keeping similar latency and area. The wider tweak input caters to both specific use cases and the design of modes of operation with higher security bounds. This is achieved through new key and tweak schedules, revised S-Box and linear layer choices, and a more comprehensive security analysis. QARMAv2 offers competitive latency and area in fully unrolled hardware implementations. Some of our results may be of independent interest. These include: new MILP models of certain classes of diffusion matrices; the comparative analysis of a full reflection cipher against an iterative half-cipher; our boomerang attack framework; and an improved approach to doubling the width of a block cipher.

IoT devices collect privacy-sensitive data, e.g., in smart grids or in medical devices, and send this data to cloud servers for further processing. In order to ensure confidentiality as well as authenticity of the sensor data in the untrusted cloud environment, we consider a transciphering scenario between embedded IoT devices and multiple cloud servers that perform secure multi-party computation (MPC). Concretely, the IoT devices encrypt their data with a lightweight symmetric cipher and send the ciphertext to the cloud servers. To obtain the secret shares of the cleartext message for further processing, the cloud servers engage in an MPC protocol to decrypt the ciphertext in a distributed manner. This way, the plaintext is never exposed to the individual servers. As an important building block in this scenario, we propose a new, provably secure family of lightweight modes for authenticated encryption with associated data (AEAD), called Eevee. The Eevee family has fully parallel decryption, making it suitable for MPC protocols for which the round complexity depends on the complexity of the function they compute. Further, our modes use the lightweight forkcipher primitive that offers fixed-length output expansion and a compact yet parallelizable internal structure. All Eevee members improve substantially over the few available state-of-the-art (SotA) MPC-friendly modes and other standard solutions. We benchmark the Eevee family on a microcontroller and in MPC. Our proposed mode Jolteon (when instantiated with ForkSkinny) provides 1.85x to 3.64x speedup in IoT-encryption time and 3x to 4.5x speedup in both MPC-decryption time and data for very short queries of 8 bytes and, 1.55x to 3.04x and 1.23x to 2.43x speedup, respectively, in MPC-decryption time and data for queries up to 500 bytes when compared against SotA MPC-friendly modes instantiated with SKINNY. We also provide two advanced modes, Umbreon and Espeon, that show a favorable performance-security trade-off with stronger security guarantees such as nonce-misuse security. Additionally, all Eevee members have full $n$-bit security (where $n$ is the block size of the underlying primitive), use a single primitive and require smaller state and HW area when compared with the SotA modes under their original security settings.

In this paper we introduce a new keyed hash function based on 32-bit integer multiplication that we call Multimixer-128. In our approach, we follow the key-then-hash parallel paradigm. So, we first add a variable length input message to a secret key and split the result into blocks. A fixed length public function based on integer multiplication is then applied on each block and their results are added to form the digest. We prove an upper bound of $2^{-127}$ for the universality of Multimixer-128 by means of the differential probability and image probability of the underlying public function. There are vector instructions for fast 32-bit integer multiplication on many CPUs and in such platforms, Multimixer-128 is very efficient. We compare our implementation of Multimixer-128 with NH hash function family that offers similar levels of security and with two fastest NIST LWC candidates. To the best of our knowledge, NH hash function is the fastest keyed hash function on software and Multimixer-128 outperforms NH while providing same levels of security.

This work studies the key-alternating ciphers (KACs) whose round permutations are not necessarily independent. We revisit existing security proofs for key-alternating ciphers with a single permutation (KACSPs), and extend their method to an arbitrary number of rounds. In particular, we propose new techniques that can significantly simplify the proofs, and also remove two unnatural restrictions in the known security bound of 3-round KACSP (Wu et al., Asiacrypt 2020). With these techniques, we prove the first tight security bound for t-round KACSP, which was an open problem. We stress that our techniques apply to all variants of KACs with non-independent round permutations, as well as to the standard KACs.

The design and analysis of dedicated tweakable block ciphers constitute a dynamic and relatively recent research field in symmetric cryptanalysis. The assessment of security in the related-tweakey model is of utmost importance owing to the existence of a public tweak. This paper proposes an automatic search model for identifying related-tweakey impossible differentials based on the propagation of states under specific constraints, which is inspired by the research of Hu et al. in ASIACRYPT 2020. Our model is universally applicable to block ciphers, but its search efficiency may be limited in some cases. To address this issue, we introduce the Locality Constraint Analysis (LCA) technique to impossible differential cryptanalysis and propose a generalized automatic search model. Technically, we transform our models into Satisfiability Modulo Theories (SMT) problems and solve them using the STP solver. We have applied our tools to several tweakable block ciphers, such as Joltik-BC, SKINNY, QARMA, and CRAFT, to evaluate their effectiveness and practicality. Specifically, we have discovered 7-round related-tweakey impossible differentials for Joltik-BC-192, and 12-round related-tweak impossible differentials, as well as 15-round related-tweakey impossible differentials for CRAFT for the first time. Based on the search results, we demonstrate that the LCA technique can be effectively performed when searching and determining the contradictory positions for the distinguisher with long trails or ciphers with large sizes in impossible differential cryptanalysis.

Various message authentication codes (MACs), including HMAC-Streebog and Streebog-K, are based on the keyless hash function Streebog. Under the assumption that the compression function of Streebog is resistant to the related key attacks, the security proofs of these algorithms were recently presented at CTCrypt 2022. We carefully detail the resources of the adversary in the related key settings, revisit the proof, and obtain tight security bounds. Let $n$ be the bit length of the hash function state. If the amount of processed data is less than about $2^{n-k}$ blocks, then for HMAC-Streebog-512 and Streebog-K, the only effective method of forgery (or distinguishing) is guessing the $k$-bit secret key or the tag if it is shorter than the key. So, we can speak about ``$k$-bit security'' without specifying the amount of material, if the key length is no longer than half of a state. The bound for HMAC-Streebog-256 is worse and equal to $2^{\frac{n}{2}-k}$ blocks.

The differential meet-in-the-middle (MITM) attack is a new cryptanalysis technique proposed at Crypto 2023 recently. It led to greatly improved attacks on round-reduced SKINNY-128-384 and AES-256. In this paper, we revisit the differential MITM attack and propose several variants by absorbing techniques widely used in the classical differential attack. In particular, we present a new differential MITM attack that generalizes the basic differential MITM attack in several aspects. As for applications, we make refinements to the 24-round attack on SKINNY-128-384; on 12-round AES-256, we show that the classical differential attack and the generalized differential MITM attack perform better than the basic differential MITM attack.

Many classical secure structures are broken by quantum attacks. Evaluating the quantum security of a structure and providing a tight security bound is a challenging research area. As a tweakable block cipher structure based on block ciphers, $\mathsf{TNT}$ was proven to have $O(2^{3n/4})$ CPA and $O(2^{n/2})$ CCA security in the classical setting. We prove that $\mathsf{TNT}$ is a quantum-secure tweakable block cipher with a bound of $O(2^{n/6})$. In addition, we show the tight quantum PRF security bound of $O(2^{n/3})$ when $\mathsf{TNT}$ is based on random functions, which is better than $O(2^{n/4})$ given by Bhaumik et al. and solves their open problem. Our proof uses the recording standard oracle with errors technique of Hosoyamada and Iwata based on Zhandry’s compressed oracle technique.

Recent works have revisited blockcipher structures to achieve MPC- and ZKP-friendly designs. In particular, Albrecht et al. (EUROCRYPT 2015) first pioneered using a novel structure SP networks with partial non-linear layers (P-SPNs) and then (ESORICS 2019) repopularized using multi-line generalized Feistel networks (GFNs). In this paper, we persist in exploring symmetric cryptographic constructions that are conducive to the applications such as MPC. In order to study the minimization of non-linearity in Type-II Generalized Feistel Networks, we generalize the (extended) GFN by replacing the bit-wise shuffle in a GFN with the stronger linear layer in P-SPN and introducing the key in each round. We call this scheme Generalized Extended Generalized Feistel Network (GEGFN). When the block-functions (or S-boxes) are public random permutations or (domain-preserving) functions, we prove CCA security for the 5-round GEGFN. Our results also hold when the block-functions are over the prime fields F_p, yielding blockcipher constructions over (F_p)^*.

Data security has become a paramount concern in the age of data driven applications, necessitating the deployment of robust encryption techniques. This paper presents an in-depth investigation into the strength and randomness of the keystream generated by the Grain cipher, a widely employed stream cipher in secure communication systems. To achieve this objective, we propose the construction of sophisticated deep learning models for keystream prediction and evaluation. The implications of this research extend to the augmentation of our comprehension of the encryption robustness offered by the Grain cipher, accomplished by harnessing the power of deep learning models for cryptanalysis. The insights garnered from this study hold significant promise for guiding the development of more resilient encryption algorithms, thereby reinforcing the security of data transmission across diverse applications.

In CRYPTO'02, Liskov et al. have introduced a new symmetric key primitive called tweakable block cipher. They have proposed two constructions of designing a tweakable block cipher from block ciphers. The first proposed construction is called $\mathsf{LRW1}$ and the second proposed construction is called $\mathsf{LRW2}$. Although, $\mathsf{LRW2}$ has been extended in later works to provide beyond birthday bound security (e.g., cascaded $\mathsf{LRW2}$ in CRYPTO'12 by Landecker et al.), but extension of the $\mathsf{LRW1}$ has received no attention until the work of Bao et al. in EUROCRYPT'20, where the authors have shown that one round extension of $\mathsf{LRW1}$, i.e., masking the output of $\mathsf{LRW1}$ with the given tweak and then re-encrypting it with the same block cipher, gives security up to $2^{2n/3}$ queries. Recently, Khairallah has shown a birthday bound distinguishing attack on the construction and hence invalidated the security claim of Bao et al. This has led to the open research question, that {\em how many round are required for cascading $\mathsf{LRW1}$ to achieve beyond birthday bound security ?} In this paper, we have shown that cascading $\mathsf{LRW1}$ up to four rounds is sufficient for ensuring beyond the birthday bound security. In particular, we have shown that $\mathsf{CLRW1}^4$ provides security up to $2^{3n/4}$ queries. Security analysis of our construction is based on the recent development of the mirror theory technique for tweakable random permutations under the framework of the Expectation Method.

In a recent paper, Khairallah demonstrated a birthday-bound attack on TNT, thereby invalidating its (beyond-the-birthday-bound) CCA security claims. In this short note, we reestablish a birthday-bound CCA security bound for TNT. Furthermore, using a minor variant of Khairallah's attack, we show that our security bound is tight. We provide a rigorous and complete attack advantage calculations to further enhance the confidence in Khairallah's proposed attack strategy.

In this paper, we present a new distinguisher for the Tweak-aNd-Tweak (TNT) tweakable block cipher with $O(2^{n/2})$ complexity. The distinguisher is an adaptive chosen ciphertext distinguisher, unlike previous attacks that are only non-adaptive chosen plaintext attacks. However, the attack contradicts the security claims made by the designers. Given TNT can be seen as the three-round CLRW1 tweakable block cipher, our attack matches its more conservative bound. We provide the distinguisher description, a probabilistic analysis of its behaviour, experimental verification and an analysis of why the proof fails to capture the security of TNT. In summary, the distinguisher is based on collision counting and exploits non-uniformity in the statistical behaviour of random permutations. It reduces the goal of finding the collision to solving a difference equation defined over a random permutation. Due to this relation, the number of collisions observed by the distinguisher is twice as expected from an ideal tweakable block cipher.