A broadcast encryption system generally involves three kinds of entities: the group manager that deals with the membership, the encryptor that encrypts the data to the registered users according to a specific policy (the target set), and the users that decrypt the data if they are authorized by the policy. Public-key broadcast encryption can be seen as removing this special role of encryptor, by allowing anybody to send encrypted data. In this paper, we go a step further in the decentralization process, by removing the group manager: the initial setup of the group, as well as the addition of further members to the system, do not require any central authority. Our construction makes black-box use of well-known primitives and can be considered as an extension to the subset-cover framework. It allows for efficient concrete instantiations, with parameter sizes that match those of the subset-cover constructions, while at the same time achieving the highest security level in the standard model under the DDH assumption.

Ciphertext policy attribute based encryption (CP-ABE) is a technique in which user with secret key containing attributes, only able to decrypt the message if the attributes in the policy match with the attributes in secret key. The existing methods that use reasonably computable decryption policies produce the ciphertext of size at least linearly varying with the number of attributes with additional pairing operations during encryption and decryption. In this paper, we propose a scheme in which ciphertext remains constant in length, irrespective of the number of attributes. Our scheme works for a threshold case: the number of attributes in a policy must be a subset of attributes in a secret key. The security of propose scheme is based on Decisional Bilinear Diffie-Hellman (DBDH) problem.

In Ciphertext-Policy Attribute Based Encryption (CP-ABE), attributes are attached to the user‟s secret key and access policy is at-tached to the ciphertext. If attributes in the secret key of a user satisfy the policy then only the genuine user can decrypt the ciphertext. How-ever, such scenario also necessitates periodic updating of the secret key with the changing attributes. According to our observations, the existing attempts at doing so are not efficient. In this paper, we propose a newer approach to add, update or delete the value of particular attribute effi-ciently without the knowledge of the other attributes.

We show that an encryption scheme cannot have a simple decryption function and be homomorphic at the same time, even with added noise. Specifically, if a scheme can homomorphically evaluate the majority function, then its decryption cannot be weakly-learnable (in particular, linear), even if large decryption error is allowed. (In contrast, without homomorphism, such schemes do exist and are presumed secure, e.g. based on LPN.) An immediate corollary is that known schemes that are based on the hardness of decoding in the presence of low hamming-weight noise cannot be fully homomorphic. This applies to known schemes such as LPN-based symmetric or public key encryption. Using these techniques, we show that the recent candidate fully homomorphic encryption, suggested by Bogdanov and Lee (ePrint '11, henceforth BL), is insecure. In fact, we show two attacks on the BL scheme: One that uses homomorphism, and another that directly attacks a component of the scheme.

This manuscript describes a proposal for a new trap-door one-way function of the multivariate-quadratic type. It was first posted to the IACR preprint server in May 2012. Subsequently, Enrico Thomae and Christopher Wolf were able to to determine that a small-minors MinRank attack works against this scheme. I would like to thank them for their close study of the proposal. The manuscript follows as originally posted, with the addition of a few references and a brief description of the successful attack (end of Section 4.1).

In a $(t,n)$ designated verifier threshold proxy signature \, scheme, an original signer can delegate his/her signing power to $n$ proxy signers such that any $t$ or more out of $n$ proxy signers can sign messages on behalf of the original signer but $t-1$ or less of the proxy signers cannot generate a valid proxy signature. Of course, the signature is issued for a designated receiver and therefore only the designated receiver can validate the proxy signature. In this paper, we propose a new designated verifier threshold proxy signature scheme and also show that the proposed scheme has provable security in the standard model. The security of proposed scheme is based on the $GBDH$ assumption and the proposed scheme satisfies all the security requirements of threshold proxy signature schemes.

We introduce the concept of \emph{Revocable Predicate Encryption (RPE)}, which extends the previous PE setting with revocation support: private keys can be used to decrypt an RPE ciphertext only if they match the decryption policy (defined via attributes encoded into the ciphertext and predicates associated with private keys) and were not revoked by the time the ciphertext was created. The first challenge in RPE schemes is to preserve privacy for RPE ciphertexts, namely to ensure the \emph{attribute-hiding} property, which is inherent to traditional PE constructions, and which implies the more basic property of payload hiding, used in the context of Attribute-Based Encryption (ABE). We formalize the notion of attribute hiding in the presence of revocation and propose our first RPE construction, called AH-RPE, which is attribute-hiding under the Decision Linear assumption in the standard model. In the AH-RPE scheme we deploy the revocation system of Lewko, Sahai, and Waters (IEEE S\&P 2010), introduced for a simpler setting of broadcast encryption, which we modify for integration with the payload-hiding ABE scheme of Okamoto and Takashima (CRYPTO 2010), after making the latter attribute-hiding by borrowing additional techniques from Lewko, Okamoto, Sahai, Takashima, and Waters (Eurocrypt 2010). As a second major step we show that RPE schemes may admit more stringent privacy requirements in comparison to PE schemes, especially when it comes to the revocation of private keys. In addition to attribute-hiding, RPE ciphertexts should ideally not leak any information about the revoked keys and by this about the revoked users. We formalize this stronger privacy notion, termed \emph{full hiding}, and propose another RPE scheme, called FH-RPE, which achieves this notion in the setting of ``sender-local revocation'' of Attrapadung and Imai (Cryptography and Coding 2009), under the same assumptions as our AH-RPE construction. Our FH-RPE scheme is also based on the attribute-hiding variant of Okamoto and Takashima's ABE scheme, yet with a different revocation method, in which we integrate the Subset-Cover Framework of Naor, Naor, and Lotspiech (CRYPTO 2001) for better efficiency.

This paper presents a new security notion, called \emph{perfect keyword privacy (PKP)}, for non-interactive public-key encryption with keyword search (PEKS) \cite{bcop04}. Although the conventional security notion for PEKS guarantees that a searchable ciphertext leaks no information about keywords, it gives no guarantee concerning leakage of a keyword from the trapdoor. PKP is a notion for overcoming this fatal deficiency. Since the trapdoor has verification functionality, the popular concept of ``indistinguishability'' is inadequate for capturing the notion of keyword privacy from the trapdoor. Hence, our formalization of PKP depends on the idea of formalizing a perfectly one-way hash function \cite{can97,cmr98}. We also present \emph{IND-PKP security} as a useful notion for showing that a given PEKS scheme has PKP. Furthermore, we present PKP+ and IND-PKP+ as enhanced notions of PKP and IND-PKP, respectively. Finally, we present several instances of an IND-PKP or IND-PKP+ secure PEKS scheme, in either the random oracle model or the standard model.

Aggregation schemes allow to combine several cryptographic values like message authentication codes or signatures into a shorter value such that, despite compression, some notion of unforgeability is preserved. Recently, Eikemeier et al. (SCN 2010) considered the notion of history-free sequential aggregation for message authentication codes, where the sequentially-executed aggregation algorithm does not need to receive the previous messages in the sequence as input. Here we discuss the idea for signatures where the new aggregate does not rely on the previous messages and public keys either, thus inhibiting the costly verifications in each aggregation step as in previous schemes by Lysyanskaya et al. (Eurocrypt 2004) and Neven (Eurocrypt 2008). Analogously to MACs we argue about new security definitions for such schemes and compare them to previous notions for history-dependent schemes. We finally give a construction based on the BLS signature scheme which satisfies our notion.

Deterministic public key encryption (D-PKE) provides an alternative to randomized public key encryption in various scenarios (e.g. search on encrypted data) where the latter exhibits inherent drawbacks. In CRYPTO'11, Brakerski and Segev formalized a framework for studying the security of deterministic public key encryption schemes with respect to auxiliary inputs. A trivial requirement is that the plaintext should not be efficiently recoverable from the auxiliary inputs. In this paper, we present an efficient deterministic public key encryption scheme in the auxiliary-input setting from lattices. The public key size, ciphertext size and ciphertext expansion factor are improved compared with the scheme proposed by Brakerski and Segev. Our scheme is also secure even in the multi-user setting where related messages may be encrypted under multiple public keys. In addition, the security of our scheme is based on the hardness of the learning with errors (LWE) problem which remains hard even for quantum algorithms. Furthermore, we consider deterministic identity-based public key encryption (D-IBE) in the auxiliary-input setting. The only known D-IBE scheme (without considering auxiliary inputs) in the standard model was proposed by Bellare et al. in EUROCRYPT'12. However, this scheme is only secure in the selective security setting, and Bellare et al. identified it as an open problem to construct adaptively secure D-IBE schemes. The second contribution of this work is to propose a D-IBE scheme from lattices that is adaptively secure.

Protocol for fair exchange of digital signatures is essential in many applications including contract signing, electronic commerce, or even peer-to-peer file sharing. In such a protocol, two parties, Alice and Bob, would like to exchange digital signatures on some messages in a fair way. It is known that a trusted arbitrator is necessary in the realization of such a protocol. We identify that in some scenarios, it is required that prior to the completion of the protocol, no observer should be able to tell whether Alice and Bob are conducting such an exchange. Consider the following scenario in which Apple engages Intel in an exchange protocol to sign a contract that terminates their OEM agreement. The information would be of value to a third party (such as the stock broker, or other OEM companies). If the protocol transcript can serve as an evidence that such a communication is in progress, any observer of this communication, including the employees of both companies, would be tempted to capture the transcript and sell it to outsiders. We introduce a new notion called \emph{perfect ambiguous optimistic fair exchange} (PAOFE), which is particularly suitable to the above scenario. PAOFE fulfils all traditional requirements of cryptographic fair exchange of digital signatures and, in addition, guarantees that the communication transcript cannot be used as a proof to convince others that the protocol is in progress. Specifically, we formalize the notion of PAOFE and present a rigorous security model in the multi-user setting under the chosen-key attack. We also present a generic construction of PAOFE from existing cryptographic primitives and prove that our proposal is secure with respect to our definition in the standard model.

In this paper, we focus on the problem of minimizing ciphertext overhead, and discuss the (im)possibility of constructing key encapsulation mechanisms (KEMs) with low ciphertext overhead. More specifically, we rule out the existence of algebraic black-box reductions from the (bounded) CCA security of a natural class of KEMs to any non-interactive problem. The class of KEMs captures the structure of the currently most efficient KEMs defined in standard prime order groups, but restricts an encapsulation to consist of a single group element and a string. This result suggests that we cannot rely on existing techniques to construct a CCA secure KEM in standard prime order groups with a ciphertext overhead lower than two group elements. Furthermore, we show how the properties of an (algebraic) programmable hash function can be exploited to construct a simple, efficient and CCA secure KEM based on the hardness of the decisional Diffie-Hellman problem with the ciphertext overhead of just a single group element. Since this KEM construction is covered by the above mentioned impossibility result, this enables us to derive a lower bound on the hash key size of an algebraic programmable hash function, and rule out the existence of algebraic $({\sf poly},n)$-programmable hash functions in prime order groups for any integer $n$. The latter result answers an open question posed by Hofheinz and Kiltz (CRYPTO'08) in the case of algebraic programmable hash functions in prime order groups.

Group signatures are a central cryptographic primitive where users can anonymously and accountably sign messages in the name of a group they belong to. Several efficient constructions with security proofs in the standard model ({\it i.e.}, without the random oracle idealization) appeared in the recent years. However, like standard PKIs, group signatures need an efficient revocation system to be practical. Despite years of research, membership revocation remains a non-trivial problem: many existing solutions do not scale well due to either high overhead or constraining operational requirements (like the need for all users to update their keys after each revocation). Only recently, Libert, Peters and Yung (Eurocrypt'12) suggested a new scalable revocation method, based on the Naor-Naor-Lotspiech (NNL) broadcast encryption framework, that interacts nicely with techniques for building group signatures in the standard model. While promising, their mechanism introduces important storage requirements at group members. Namely, membership certificates, which used to have constant size in existing standard model constructions, now have polylog size in the maximal cardinality of the group (NNL, after all, is a tree-based technique and such dependency is naturally expected). In this paper we show how to obtain private keys of {\it constant} size. To this end, we introduce a new technique to leverage the NNL subset cover framework in the context of group signatures but, perhaps surprisingly, without logarithmic relationship between the size of private keys and the group cardinality. Namely, we provide a way for users to efficiently prove their membership of one of the generic subsets in the NNL subset cover framework. This technique makes our revocable group signatures competitive with ordinary group signatures ({\it i.e.}, without revocation) in the standard model. Moreover, unrevoked members (as in PKIs) still do not need to update their keys at each revocation.

Group signatures are a central cryptographic primitive, simultaneously supporting accountability and anonymity. They allow users to anonymously sign messages on behalf of a group they are members of. The recent years saw the appearance of several constructions with security proofs in the standard model ({\it i.e.}, without appealing to the random oracle heuristic). For a digital signature scheme to be adopted, an efficient revocation scheme (as in regular PKI) is absolutely necessary. Despite over a decade of extensive research, membership revocation remains a non-trivial problem in group signatures: all existing solutions are not truly scalable due to either high overhead (e.g., large group public key size), or limiting operational requirement (the need for all users to follow the system's entire history). In the standard model, the situation is even worse as many existing solutions are not readily adaptable. To fill this gap and tackle this challenge, we describe a new revocation approach based, perhaps somewhat unexpectedly, on the Naor-Naor-Lotspiech framework which was introduced for a different problem (namely, that of broadcast encryption). Our mechanism yields efficient and scalable revocable group signatures in the standard model. In particular, the size of signatures and the verification cost are independent of the number of revocations and the maximal cardinality $N$ of the group while other complexities are at most polylogarithmic in $N$. Moreover, the schemes are history-independent: unrevoked group members do not have to update their keys when a revocation occurs.

Verified security provides a firm foundation for cryptographic proofs by means of rigorous programming language techniques and verification methods. EasyCrypt is a framework that realizes the verified security paradigm and supports the machine-checked construction and verification of cryptographic proofs using state-of-the-art SMT solvers, automated theorem provers and interactive proof assistants. Previous experiments have shown that EasyCrypt is effective for a posteriori validation of cryptographic systems. In this paper, we report on the first application of verified security to a novel cryptographic construction, with strong security properties and interesting practical features. Specifically, we use EasyCrypt to prove the IND-CCA security of a redundancy-free public-key encryption scheme based on trapdoor one-way permutations. Somewhat surprisingly, we show that even with a zero-length redundancy, Boneh's SAEP scheme (an OAEP-like construction with a single-round Feistel network rather than two) converts a trapdoor one-way permutation into an IND-CCA-secure scheme, provided the permutation satisfies two additional properties. We then prove that the Rabin function and RSA with short exponent enjoy these properties, and thus can be used to instantiate the construction we propose to obtain efficient encryption schemes. The reduction that justifies the security of our construction is tight enough to achieve practical security with reasonable key sizes.

Motivated by the question of access control in cloud storage, we consider the problem using Attribute-Based Encryption (ABE) in a setting where users' credentials may change and ciphertexts may be stored by a third party. We find that a comprehensive solution to our problem must simultaneously allow for the revocation of ABE private keys as well as allow for the ability to update ciphertexts to reflect the most recent updates. Our main result is obtained by pairing two contributions: - Revocable Storage. We ask how a third party can process a ciphertext to disqualify revoked users from accessing data that was encrypted in the past, while the user still had access. In applications, such storage may be with an untrusted entity and as such, we require that the ciphertext management operations can be done without access to any sensitive data (which rules out decryption and re-encryption). We define the problem of revocable storage and provide a fully secure construction. Our core tool is a new procedure that we call ciphertext delegation. One can apply ciphertext delegation on a ciphertext encrypted under a certain access policy to `re-encrypt' it to a more restrictive policy using only public information. We provide a full analysis of the types of delegation possible in a number of existing ABE schemes. - Protecting Newly Encrypted Data. We consider the problem of ensuring that newly encrypted data is not decryptable by a user's key if that user's access has been revoked. We give the first method for obtaining this revocation property in a fully secure ABE scheme. We provide a new and simpler approach to this problem that has minimal modifications to standard ABE. We identify and define a simple property called piecewise key generation which gives rise to efficient revocation. We build such solutions for Key-Policy and Ciphertext-Policy Attribute-Based Encryption by modifying an existing ABE scheme due to Lewko et al. to satisfy our piecewise property and prove security in the standard model. It is the combination of our two results that gives an approach for revocation. A storage server can update stored ciphertexts to disqualify revoked users from accessing data that was encrypted before the user's access was revoked. This is the full version of the Crypto 2012 paper.

We identify a potential weakness in the standard security model for dynamic group signatures which appears to have been overlooked previously. More specifically, we highlight that even if a scheme provably meets the security requirements of the model, a malicious group member can potentially claim ownership of a group signature produced by an honest group member by forging a proof of ownership. This property leads to a number of vulnerabilities in scenarios in which dynamic group signatures are likely to be used. We furthermore show that the dynamic group signature scheme by Groth (ASIACRYPT 2007) does not provide protection against this type of malicious behavior. To address this, we introduce the notion of \emph{opening soundness} for group signatures which essentially requires that it is infeasible to produce a proof of ownership of a valid group signature for any user except the original signer. We then show a relatively simple modification of the scheme by Groth which allows us to prove opening soundness for the modified scheme without introducing any additional assumptions. We believe that opening soundness is an important and natural security requirement for group signatures, and hope that future schemes will adopt this type of security.

Genus 2 curves with simple but not absolutely simple jacobians can be used to construct pairing-based cryptosystems more efficient than for a generic genus 2 curve. We show that there is a full analogy between methods for constructing ordinary pairing-friendly elliptic curves and simple abelian varieties, which are iogenous over some extension to a product of elliptic curves. We extend the notion of complete, complete with variable discriminant, and sparse families introduced in by Freeman, Scott and Teske for elliptic curves, and we generalize the Cocks-Pinch method and the Brezing-Weng method to construct families of each type. To realize abelian surfaces as jacobians we use of genus 2 curves of the form $y^2=x^5+ax^3+bx$ or $y^2=x^6+ax^3+b$, and apply the method of Freeman and Satoh. As applications we find some families of abelian surfaces with recorded $\rho$-value $\rho=2$ for embedding degrees $k=3,4,6,12$, or $\rho=2.1$ for $k=27,54$. We also give variable-discriminant families with best $\rho$-values.

We present the notion of PROG-KDM security for public-key encryption schemes. This security notion captures both KDM security and revealing of secret keys (key corruptions) in a single definition. This is achieved by requiring the existence of a simulator that can program ciphertexts when a secret key is revealed, i.e., the simulator can delay the decision what plaintext is contained in what ciphertext to the moment where the ciphertext is opened. The definition is formulated in the random oracle model. We show that PROG-KDM security can be achieved by showing that a natural and practical construction in the ideal cipher model is PROG-KDM secure (hybrid encryption using authenticated CBC encryption).

In this paper, we present two non-zero inner-product encryption (NIPE) schemes that are adaptively secure under a standard assumption, the decisional linear (DLIN) assumption, in the standard model. One of the proposed NIPE schemes features constant-size ciphertexts and the other features constant-size secret-keys. Our NIPE schemes imply an identity-based revocation (IBR) system with constant-size ciphertexts or constant-size secret-keys that is adaptively secure under the DLIN assumption. Any previous IBR scheme with constant-size ciphertexts or constant-size secret-keys was not adaptively secure in the standard model. This paper also presents two zero inner-product encryption (ZIPE) schemes each of which has constant-size ciphertexts or constant-size secret-keys and is adaptively secure under the DLIN assumption in the standard model. They imply an identity-based broadcast encryption (IBBE) system with constant-size ciphertexts or constant-size secret-keys that is adaptively secure under the DLIN assumption. We also extend the proposed ZIPE schemes into two directions, one is a fully-attribute-hiding ZIPE scheme with constant-size secret-keys, and the other a hierarchical ZIPE scheme with constant-size ciphertexts.

In this paper, we present the first decentralized multi-authority attribute-based signature (DMA-ABS) scheme, in which no central authority and no trusted setup are required. The proposed DMA-ABS scheme for general (non-monotone) predicates is fully secure(adaptive-predicate unforgeable and perfect private) under a standard assumption, the decisional linear (DLIN) assumption, in the random oracle model. Our DMA-ABS scheme is comparably as efficient as the most efficient ABS scheme. As a by-product, this paper also presents an adaptively secure DMA functional encryption (DMA-FE) scheme under the DLIN assumption.

We introduce the notion of Cross-Unlinkability for group signature schemes. Considering groups organized in a tree structure, where belonging to the parent group is required to join a new group, Cross-Unlinkability enables a cascade revocation process that takes into account the underlying tree structure, while ensuring anonymity for non-revoked users, in particular, towards the managers of the other groups. We show how to achieve Cross-Unlinkability using the Verifier-Local Revocation group signature scheme of Bringer and Patey at Secrypt 2012, by exploiting its property of Backward Unlinkability.

Secrecy of decryption keys is an important pre-requisite for security of any encryption scheme and compromised private keys must be immediately replaced. \emph{Forward Security (FS)}, introduced to Public Key Encryption (PKE) by Canetti, Halevi, and Katz (Eurocrypt 2003), reduces damage from compromised keys by guaranteeing confidentiality of messages that were encrypted prior to the compromise event. The FS property was also shown to be achievable in (Hierarchical) Identity-Based Encryption (HIBE) by Yao, Fazio, Dodis, and Lysyanskaya (ACM CCS 2004). Yet, for emerging encryption techniques, offering flexible access control to encrypted data, by means of functional relationships between ciphertexts and decryption keys, FS protection was not known to exist.\smallskip In this paper we introduce FS to the powerful setting of \emph{Hierarchical Predicate Encryption (HPE)}, proposed by Okamoto and Takashima (Asiacrypt 2009). Anticipated applications of FS-HPE schemes can be found in searchable encryption and in fully private communication. Considering the dependencies amongst the concepts, our FS-HPE scheme implies forward-secure flavors of Predicate Encryption and (Hierarchical) Attribute-Based Encryption.\smallskip Our FS-HPE scheme guarantees forward security for plaintexts and for attributes that are hidden in HPE ciphertexts. It further allows delegation of decrypting abilities at any point in time, independent of FS time evolution. It realizes zero-inner-product predicates and is proven adaptively secure under standard assumptions. As the ``cross-product" approach taken in FS-HIBE is not directly applicable to the HPE setting, our construction resorts to techniques that are specific to existing HPE schemes and extends them with what can be seen as a reminiscent of binary tree encryption from FS-PKE.

We consider designing broadcast encryption schemes with constant-size secret keys and ciphertexts, achieving chosen-ciphertext security. We first argue that known CPA-to-CCA transforms currently do not yield such schemes. We then propose a scheme, modifying a previous selective CPA secure proposal by Boneh, Gentry, and Waters. Our proposed scheme has constant-size secret keys and ciphertexts and we prove that it is selective chosen-ciphertext secure based on standard assumptions. Our scheme has ciphertexts that are shorter than those of the previous CCA secure proposals. Then we propose a second scheme that provides the functionality of both broadcast encryption and revocation schemes simultaneously using the same set of parameters. Finally we show that it is possible to prove our first scheme adaptive chosen-ciphertext secure under reasonable extensions of the bilinear Diffie-Hellman exponent and the knowledge of exponent assumptions. We prove both of these extended assumptions in the generic group model. Hence, our scheme becomes the first to achieve constant-size secret keys and ciphertexts (both asymptotically optimal) and adaptive chosen-ciphertext security at the same time.

Designing an ID based signcryption scheme in the standard model is among the most interesting and important problems in cryptography. However, all the existing systems in the ID based setting, in the standard model, do not have either the unforgeability property or the indistinguishability property or both of them. In this paper, we present the first provably secure ID based signcryption scheme in the standard model with both these properties. The unforgeability property of this scheme is based on the hardness of Computational Diffie-Hellman problem and the indistinguishability property of this scheme is based on the hardness of Decisional Bilinear Diffie-Hellman problem. Our scheme is strongly unforgeable in the strong attack mode called insider security. Moreover, our scheme possess an interesting property called public verifiability of the ciphertext. Our scheme integrates cleverly, a modified version of Waters' IBE and a suitably modified version of the ID based signature scheme in the standard model proposed by Paterson et al. However, our security reductions are more efficient. Specifically, while the security reductions for indistinguishability is similar to the bounds of Waters' scheme, the unforgeability reductions are way better than the bounds for Paterson et al.'s scheme.

At EUROCRYPT '10, van Dijk, Gentry, Halevi and Vaikuntanathan presented simple fully-homomorphic encryption (FHE) schemes based on the hardness of approximate integer common divisors problems, which were introduced in 2001 by Howgrave-Graham. There are two versions for these problems: the partial version (PACD) and the general version (GACD). The seemingly easier problem PACD was recently used by Coron, Mandal, Naccache and Tibouchi at CRYPTO '11 to build a more efficient variant of the FHE scheme by van Dijk {\em et al.}. We present a new PACD algorithm whose running time is essentially the ``square root'' of that of exhaustive search, which was the best attack in practice. This allows us to experimentally break the FHE challenges proposed by Coron {\em et al.} Our PACD algorithm directly gives rise to a new GACD algorithm, which is exponentially faster than exhaustive search: namely, the running time is essentially the $3/4$-th root of that of exhaustive search. Interestingly, our main technique can also be applied to other settings, such as noisy factoring and attacking low-exponent RSA.

The Rainbow Signature Scheme is a non-trivial generalization of the well known Unbalanced Oil and Vinegar Signature Scheme (Eurocrypt '99) minimizing the length of the signatures. Recently a new variant based on non-commutative rings, called NC-Rainbow, was introduced at CT-RSA 2012 to further minimize the secret key size. We disprove the claim that NC-Rainbow is as secure as Rainbow in general and show how to reduce the complexity of MinRank attacks from 2^288 to 2^112 and of HighRank attacks from 2^128 to 2^96 for the proposed instantiation over the ring of Quaternions. We further reveal some facts about Quaternions that increase the complexity of the signing algorithm. We show that NC-Rainbow is just a special case of introducing further structure to the secret key in order to decrease the key size. As the results are comparable with the ones achieved by equivalent keys, which provably do not decrease security, and far worse than just using a PRNG, we recommend not to use NC-Rainbow.

We present a generalized learning with errors (LWE) problem, which is essentially a simple and direct extension of the original LWE problem to the case of matrices. Then we use this new version of LWE problem, which we call matrix LWE problem to build new cryptographic schemes, which include a new scalable key distribution scheme, a new key exchanges scheme and a new simple identity-based encryption scheme.

We provide a functional encryption system that supports functionality for regular languages. In our system a secret key is associated with a Deterministic Finite Automata (DFA) M. A ciphertext, CT, encrypts a message m and is associated with an arbitrary length string w. A user is able to decrypt the ciphertext CT if and only if the DFA M associated with his private key accepts the string w. Compared with other known functional encryption systems, this is the first system where the functionality is capable of recognizing an unbounded language. For example, in (Key-Policy) Attribute-Based Encryption (ABE) a private key SK is associated with a single boolean formula which operates over a fixed number of boolean variables from the ciphertext. In contrast, in our system a DFA M will meaningfully operate over an arbitrary length input w. We propose a system that utilizes bilinear groups. Our solution is a "public index" system, where the message m is hidden, but the string w is not. We prove security in the selective model under a variant of the decision l-Bilinear Diffie-Hellman Exponent (BDHE) assumption that we call the decision l-Expanded BDHE problem.

We extend the work of Bellare, Boldyreva and Staddon on the systematic analysis of randomness reuse to construct multi-recipient encryption schemes to the case where randomness is reused across different cryptographic primitives. We find that through the additional binding introduced through randomness reuse, one can actually obtain a security amplification with respect to the standard black-box compositions, and achieve a stronger level of security. We introduce stronger notions of security for encryption and signatures, where challenge messages can depend in a restricted way on the random coins used in encryption, and show that two variants of the KEM/DEM paradigm give rise to encryption schemes that meet this enhanced notion of security. We obtain a very efficient signcryption scheme that is secure against insider attackers without random oracles.

Pollard's rho algorithm, along with parallelized, vectorized, and negating variants, is the standard method to compute discrete logarithms in generic prime-order groups. This paper presents two reasons that Pollard's rho algorithm is farther from optimality than generally believed. First, ``higher-degree local anti-collisions'' make the rho walk less random than the predictions made by the conventional Brent--Pollard heuristic. Second, even a truly random walk is suboptimal, because it suffers from ``global anti-collisions'' that can at least partially be avoided. For example, after (1.5+o(1))\sqrt(l) additions in a group of order l (without fast negation), the baby-step-giant-step method has probability 0.5625+o(1) of finding a uniform random discrete logarithm; a truly random walk would have probability 0.6753\ldots+o(1); and this paper's new two-grumpy-giants-and-a-baby method has probability 0.71875+o(1).

We give details of the factorization of RSA-704 with CADO-NFS. This is a record computation with publicly available software tools. The aim of this experiment was to stress CADO-NFS - which was originally designed for 512-bit factorizations - for larger inputs, and to identify possible rooms of improvement.

In 1989, Adi Shamir proposed a new zero-knowledge identification scheme based on a NP-complete problem called PKP for Permuted Kernel Problem. For a given prime p, a given matrix A and a given vector V , the problem is to find a permutation \pi such that the permuted vector V_\pi verifies A.V_\pi = 0 mod p. This scheme is still in 2011 known as one of the most efficient identification scheme based on a combinatorial problem. However, we will see in this paper that it is possible to improve this scheme significantly by combining new ideas in order to reduce the total number of computations to be performed and to improve very efficiently the security against side channel attacks using precomputations. We will obtain like this a new scheme that we have called SPKP. Moreover, if we use precomputed values in the scheme SPKP, then the prover will need to perform no computations (i.e. only selection and transmission of precomputed values). This is very interesting for security against side channel attacks because our scheme is zero-knowledge and we don't perform any computations using the key during the identification so we prove that any attacker (even using side channel attacks) being successfully identified implies that he has a solution to the NP-complete problem PKP.

We present new candidates for quantum-resistant public-key cryptosystems based on the conjectured difficulty of finding isogenies between supersingular elliptic curves. The main technical idea in our scheme is that we transmit the images of torsion bases under the isogeny in order to allow the parties to construct a shared commutative square despite the noncommutativity of the endomorphism ring. Our work is motivated by the recent development of a subexponential-time quantum algorithm for constructing isogenies between ordinary elliptic curves. In the supersingular case, by contrast, the fastest known quantum attack remains exponential, since the noncommutativity of the endomorphism ring means that the approach used in the ordinary case does not apply. We give a precise formulation of the necessary computational assumptions along with a discussion of their validity, and prove the security of our protocols under these assumptions. In addition, we present implementation results showing that our protocols are multiple orders of magnitude faster than previous isogeny-based cryptosystems over ordinary curves. This paper is an extended version of~\cite{pqcrypto}. We add a new zero-knowledge identification scheme, and detailed security proofs for the protocols. We also present a new, asymptotically faster, algorithm for key generation, a thorough study of its optimization, and new experimental data.

This paper presents efficient structure-preserving signature schemes based on assumptions as simple as Decision-Linear. We first give two general frameworks for constructing fully secure signature schemes from weaker building blocks such as two-tier signatures and random-message secure signatures. They can be seen as refinements of the Even-Goldreich-Micali framework, and preserve many desirable properties of the underlying schemes such as constant signature size and structure preservation. We then instantiate them based on simple (i.e., not q-type) assumptions over symmetric and asymmetric bilinear groups. The resulting schemes are structure-preserving and yield constant-size signatures consisting of 11 to 17 group elements, which compares favorably to existing schemes relying on q-type assumptions for their security.

At Eurocrypt 2010, Freeman proposed a transformation from pairing-based schemes in composite-order bilinear groups to equivalent ones in prime-order bilinear groups. His transformation can be applied to pairing-based cryptosystems exploiting only one of two properties of composite-order bilinear groups: cancelling and projecting. At Asiacrypt 2010, Meiklejohn, Shacham, and Freeman showed that prime-order bilinear groups according to Freeman's construction cannot have two properties simultaneously except negligible probability and, as an instance of implausible conversion, proposed a (partially) blind signature scheme whose security proof exploits both the cancelling and projecting properties of composite-order bilinear groups. In this paper, we invalidate their evidence by presenting a security proof of the prime-order version of their blind signature scheme. Our security proof follows a different strategy and exploits only the projecting property. Instead of the cancelling property, a new property, that we call {\em translating}, on prime-order bilinear groups plays an important role in the security proof, whose existence was not known in composite-order bilinear groups. With this proof, we obtain a $2$-move (i.e., round optimal) (partially) blind signature scheme (without random oracle) based on the decisional linear assumption in the common reference string model, which is of independent interest. As the second contribution of this paper, we construct prime-order bilinear groups that possess both the cancelling and projecting properties at the same time by considering more general base groups. That is, we take a rank $n$ $\zp$-submodule of $\zp^{n^2}$, instead of $\zp^n$, to be a base group $G$, and consider the projections into its rank 1 submodules. We show that the subgroup decision assumption on this base group $G$ holds in the generic bilinear group model for $n=2$, and provide an efficient membership-checking algorithm to $G$, which was trivial in the previous setting. Consequently, it is still open whether there exists a cryptosystem on composite-order bilinear groups that cannot be constructed on prime-order bilinear groups.

There are many useful cryptographic schemes, such as ID-based encryption, short signature, keyword searchable encryption, attribute-based encryption, functional encryption, that use a bilinear pairing. It is important to estimate the security of such pairing-based cryptosystems in cryptography. The most essential number-theoretic problem in pairing-based cryptosystems is the discrete logarithm problem (DLP) because pairing-based cryptosystems are no longer secure once the underlining DLP is broken. One efficient bilinear pairing is the $\eta_T$ pairing defined over a supersingular elliptic curve $E$ on the finite field $GF(3^n)$ for a positive integer $n$. The embedding degree of the $\eta_T$ pairing is $6$; thus, we can reduce the DLP over $E$ on $GF(3^n)$ to that over the finite field $GF(3^{6n})$. In this paper, for breaking the $\eta_T$ pairing over $GF(3^n)$, we discuss solving the DLP over $GF(3^{6n})$ by using the function field sieve (FFS), which is the asymptotically fastest algorithm for solving a DLP over finite fields of small characteristics. We chose the extension degree $n=97$ because it has been intensively used in benchmarking tests for the implementation of the $\eta_T$ pairing, and the order (923-bit) of $GF(3^{6\cdot 97})$ is substantially larger than the previous world record (676-bit) of solving the DLP by using the FFS. We implemented the FFS for the medium prime case (JL06-FFS), and propose several improvements of the FFS, for example, the lattice sieve for JL06-FFS and the filtering adjusted to the Galois action. Finally, we succeeded in solving the DLP over $GF(3^{6\cdot 97})$. The entire computational time of our improved FFS requires about 148.2 days using 252 CPU cores. Our computational results contribute to the secure use of pairing-based cryptosystems with the $\eta_T$ pairing.

In this paper, we present a new class of knapsack type PKC referred to as K(II)$\Sigma\Pi$PKC. In K(II)$\Sigma\Pi$PKC, Bob randomly constructs a very small subset of Alice's set of public key whose order is very large, under the condition that the coding rate $\rho$ satisfies $0.01 < \rho < 0.5$. In K(II)$\Sigma\Pi$PKC, no secret sequence such as super-increasing sequence or shifted-odd sequence but the sequence whose component is constructed by a product of the same number of many prime numbers of the same size, is used. We show that K(II)$\Sigma\Pi$PKC is secure against the attacks such as LLL algorithm, Shamir's attack etc. , because a subset of Alice's public keys is chosen entirely in a probabilistic manner at the sending end. We also show that K(II)$\Sigma\Pi$PKC can be used as a member of the class of common key cryptosystems because the list of the subset randomly chosen by Bob can be used as a common key between Bob and Alice, provided that the conditions given in this paper are strictly observed, without notifying Alice of his secret key through a particular secret channel.

In this paper, we propose an efficient, standard model, semi-generic transformation of selective-secure (Hierarchical) Identity-Based Encryption schemes into fully secure ones. The main step is a procedure that uses admissible hash functions (whose existence is implied by collision-resistant hash functions) to convert any selective-secure \emph{wildcarded} identity-based encryption (WIBE) scheme into a fully secure (H)IBE scheme. Since building a selective-secure WIBE, especially with a selective-secure HIBE already in hand, is usually much less involved than directly building a fully secure HIBE, this transform already significantly simplifies the latter task. This black-box transformation easily extends to schemes secure in the Continual Memory Leakage (CML) model of Brakerski et al. (FOCS 2010), which allows us obtain a new fully secure IBE in that model. We furthermore show that if a selective-secure HIBE scheme satisfies a particular security notion, then it can be generically transformed into a selective-secure WIBE. We demonstrate that several current schemes already fit this new definition, while some others that do not obviously satisfy it can still be easily modified into a selective-secure WIBE.

Up to now, several non-delegatable identity-based (strong) designated verifier signature schemes using bilinear pairings are proposed. In these identity-based (strong) designated verifier signature schemes, bilinear pairings are employed either in signing and verifying steps or only in the verifying step. However, the computation cost of pairings at a security level equivalent to a 128-bit symmetric key of AES is approximately 20 times higher than that of exponentiation over an elliptic curve group. Hence, presenting a (strong) designated verifier signature scheme which is identity-based without pairings and supports non-delegatability as well is vital. In this study, a non-delegatable identity-based designated verifier signature scheme without bilinear pairings using two concatenated Schnorr signatures is proposed. Our construction not only is approximately 40 times more efficient compared to the existing non-delegatable identity-based (strong) designated verifier signature schemes due to the avoiding bilinear pairings but also it is provable secure in the random oracle.

The security of pairing-based cryptosystems depends on the difficulty of the discrete logarithm problem (DLP) over certain types of finite fields. One of the most efficient algorithms for computing a pairing is the $\eta_T$ pairing over supersingular curves on finite fields whose characteristic is $3$. Indeed many high-speed implementations of this pairing have been reported, and it is an attractive candidate for practical deployment of pairing-based cryptosystems. The embedding degree of the $\eta_T$ pairing is 6, so we deal with the difficulty of a DLP over the finite field $ GF(3^{6n})$, where the function field sieve (FFS) is known as the asymptotically fastest algorithm of solving it. Moreover, several efficient algorithms are employed for implementation of the FFS, such as the large prime variation. In this paper, we estimate the time complexity of solving the DLP for the extension degrees $n=97,163, 193,239,313,353,509$, when we use the improved FFS. To accomplish our aim, we present several new computable estimation formulas to compute the explicit number of special polynomials used in the improved FFS. Our estimation contributes to the evaluation for the key length of pairing-based cryptosystems using the $\eta_T$ pairing.

Elliptic curve cryptography (ECC) is an efficient public cryptosystem with a short key size. For this reason it is suitable for implementing on memory-constraint devices such as smart cards, mobile devices, etc. However, these devices leak information about their private key through side channels (power consumption, electromagnetic radiation, timing etc) during cryptographic processing. In this paper we have examined countermeasures against a specific class of side channel attacks (power analysis) called Zero-Value Point Attack (ZVP), using elliptic curve isomorphism and isogeny. We found that these methods are an efficient way of securing cryptographic devices using ECC against ZVP attack. Our main contribution is to extend the work of Akishita and Takagi [3,2] to binary fields. We also provide a more detail analysis of the ZVP attack over prime fields.

We propose a new multivariate probabilistic encryption scheme with decryption errors MQQ-ENC that belongs to the family of MQQ-based public key schemes. Similarly to MQQ-SIG, the trapdoor is constructed using quasigroup string transformations with multivariate quadratic quasigroups, and a minus modifier with relatively small and fixed number of removed equations. To make the decryption possible and also efficient, we use a universal hash function to eliminate possibly wrong plaintext candidates. We show that, in this way, the probability of erroneous decryption becomes negligible. MQQ-ENC is defined over the fields $\mathbb{F}_{2^k}$ for any $k \geq 1$, and can easily be extended to any $\mathbb{F}_{p^k}$, for prime $p$. One important difference from MQQ-SIG is that in MQQ-ENC we use left MQQs (LMQQs) instead of bilinear MQQs. Our choice can be justified by our extensive experimental analysis that showed the superiority of the LMQQs over the bilinear MQQs for the design of MQQ-ENC. We apply the standard cryptanalytic techniques on MQQ-ENC, and from the results, we pose a plausible conjecture that the instances of the MQQ-ENC trapdoor are hard instances with respect to the MQ problem. Under this assumption, we adapt the Kobara-Imai conversion of the McEliece scheme for MQQ-ENC and prove that it provides $\mathsf{IND-CCA}$ security despite the negligible probability of decryption errors. We also recommend concrete parameters for MQQ-ENC for encryption of blocks of 128 bits for a security level of $\mathcal{O}(2^{128})$.

Bergman's Ring $E_p$, parameterized by a prime number $p$, is a ring with $p^5$ elements that cannot be embedded in a ring of matrices over any commutative ring. This ring was discovered in 1974. In 2011, Climent, Navarro and Tortosa described an efficient implementation of $E_p$ using simple modular arithmetic, and suggested that this ring may be a useful source for intractable cryptographic problems. We present a deterministic polynomial time reduction of the Discrete Logarithm Problem in $E_p$ to the classical Discrete Logarithm Problem in $\Zp$, the $p$-element field. In particular, the Discrete Logarithm Problem in $E_p$ can be solved, by conventional computers, in sub-exponential time. Along the way, we collect a number of useful basic reductions for the toolbox of discrete logarithm solvers.

Fully homomorphic encryption (FHE) provides a simple template for secure computation between two parties (Alice and Bob) where: (I) Alice encrypts her input under her key, (II) Bob homomorphically evaluates the desired function on Alice's ciphertext and his own input, and sends the encrypted output to Alice. Extending this approach to multiple parties raises the problem of which key to encrypt under; if all parties choose a key on their own, then homomorphic evaluation on ciphertexts under different keys will not be possible, and if a single party chooses the key for everyone then corrupting this party will break privacy for all. In this work, we explore the option of using threshold fully homomorphic encryption (TFHE), allowing many parties to cooperatively generate a common public key whose secret key is shared/distributed among them. Moreover, the parties can cooperatively decrypt a ciphertext without learning anything but the plaintext. We show how to instantiate this approach efficiently using the recent FHE schemes of Brakerski et al. (FOCS '11, ITCS '12) based on the learning with errors (LWE) assumption. Our main tool is to exploit the property that such LWE-based encryption schemes are homomorphic over their keys. Using TFHE, we construct multiparty computation (MPC) protocols secure against fully malicious settings, tolerating any number of corruptions, and providing security in the universal composability framework. Our schemes have several benefits fits over prior templates for MPC. Interaction: We get protocols with only 3 rounds of interaction in the common random string model, or 2 rounds with a reusable public-key infrastructure, improving on prior known results. Communication: The communication in our protocol is only proportional to the input and output size of the function being evaluated and independent of its circuit size. Computation: The only computation that depends on the size of the circuit being computed is a homomorphic evaluation over public ciphertexts. This computation can be performed by a single party or can be outsourced to an external server. Novel Approach: Prior approaches to MPC with a dishonest majority rely in part on some combination of the techniques of Yao (FOCS '86) and/or Goldreich, Micali and Wigderson (STOC '87). Our approach is fundamentally different and relies only on the homomorphic properties of LWE-based encryption.

The Ring-LWE problem, introduced by Lyubashevsky, Peikert, and Regev (Eurocrypt 2010), has been steadily finding many uses in numerous cryptographic applications. Still, the Ring-LWE problem defined in [LPR10] involves the fractional ideal $R^\vee$, the dual of the ring $R$, which is the source of many theoretical and implementation technicalities. Until now, getting rid of $R^\vee$, required some relatively complex transformation that substantially increase the magnitude of the error polynomial and the practical complexity to sample it. It is only for rings $R=\Z[X]/(X^n+1)$ where $n$ a power of $2$, that this transformation is simple and benign. In this work we show that by applying a different, and much simpler transformation, one can transfer the results from [LPR10] into an ``easy-to-use'' Ring-LWE setting ({\em i.e.} without the dual ring $R^\vee$), with only a very slight increase in the magnitude of the noise coefficients. Additionally, we show that creating the correct noise distribution can also be simplified by generating a Gaussian distribution over a particular extension ring of $R$, and then performing a reduction modulo $f(X)$. In essence, our results show that one does not need to resort to using any algebraic structure that is more complicated than polynomial rings in order to fully utilize the hardness of the Ring-LWE problem as a building block for cryptographic applications.

In many applications of group signatures, not only a signer's identity but also which group the signer belongs to is sensitive information regarding signer privacy. In this paper, we study these applications and combine a group signature with a ring signature to create a ring group signature, which specifies a set of possible groups without revealing which member of which group produced the signature. The main contributions of this paper are a formal definition of a ring group signature scheme and its security model, a generic construction and a concrete example of such a scheme. Both the construction and concrete scheme are provably secure if the underlying group signature and ring signature schemes are secure.

In this work, we study a new multivariate quadratic (MQ) assumption that can be used to construct public-key encryption schemes. In particular, we research in the following two directions: We establish a precise \emph{asymptotic} formulation of a family of hard MQ problems, and provide empirical evidence to confirm the hardness. %Since there are many practical solvers studied and implemented during the studies of algebraic attacks, we use We construct public-key encryption schemes, and prove their security under the hardness assumption of this family. Also, we provide a new \emph{perspective} to look at MQ systems that plays a key role to our design and proof of security. As a consequence, we construct the \emph{first} public-key encryption scheme that is \emph{provably secure} under the MQ assumption. Moreover, our public-key encryption scheme is efficient in the sense that it only needs a ciphertext length $L + \poly(k)$ to encrypt a message $M\in \{0, 1 \}^{L}$ for any un-prespecified polynomial $L$, where $k$ is the security parameter. This is essentially \emph{optimal} since an additive overhead is the best we can hope for.

A self-pairing is a pairing computation where both inputs are the same group element. Self-pairings are used in some cryptographic schemes and protocols. In this paper, we show how to compute the Tate-Lichtenbaum pairing (D,\phi(D)) on a curve more efficiently than the general case. The speedup is obtained by requiring a simpler final exponentiation. We also discuss how to use this pairing in cryptographic applications.

We present a new tensoring technique for LWE-based fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically ($B \to B^2\cdot\poly(n)$) with every multiplication (before ``refreshing''), our noise only grows linearly ($B \to B\cdot\poly(n)$). We use this technique to construct a \emph{scale-invariant} fully homomorphic encryption scheme, whose properties only depend on the ratio between the modulus $q$ and the initial noise level $B$, and not on their absolute values. Our scheme has a number of advantages over previous candidates: It uses the same modulus throughout the evaluation process (no need for ``modulus switching''), and this modulus can take arbitrary form, including a power of $2$ which carries obvious advantages for implementation. In addition, security can be \emph{classically} reduced to the worst-case hardness of the GapSVP problem (with quasi-polynomial approximation factor), whereas previous constructions could only exhibit a quantum reduction to GapSVP.

The use of (hyper)elliptic curves in cryptography relies on the ability to compute the Jacobian order of a given curve. Recently, Satoh proposed a probabilistic polynomial time algorithm to test whether the Jacobian -- over a finite field $\mathbb{F}_q$ -- of a hyperelliptic curve of the form $Y^2 = X^5 + aX^3 + bX$ (with $a,b \in \mathbb{F}_q^*$) has a large prime factor. His approach is to obtain candidates for the zeta function of the Jacobian over $\mathbb{F}_q^*$ from its zeta function over an extension field where the Jacobian splits. We extend and generalize Satoh's idea to provide \emph{explicit} formulas for the zeta function of the Jacobian of genus 2 hyperelliptic curves of the form $Y^2 = X^5 + aX^3 + bX$ and $Y^2 = X^6 + aX^3 + b$ (with $a,b \in \mathbb{F}_q^*$). Our results are proved by elementary (but intricate) polynomial root-finding techniques. Hyperelliptic curves with small embedding degree and large prime-order subgroup are key ingredients for implementing pairing-based cryptographic systems. Using our closed formulas for the Jacobian order, we present several algorithms to obtain so-called \emph{pairing-friendly} genus 2 hyperelliptic curves. Our method relies on techniques initially proposed to produce pairing-friendly elliptic curves (namely, the Cocks-Pinch method and the Brezing-Weng method). We demonstrate this method by constructing several interesting curves with $\rho$-values around 3. We found for each embedding degree $5 \leqslant k \leqslant 35$ a family of curves of $\rho$-value between $2.25$ and $4$.

In this study, a novel pairing based strong designated verifier signature scheme based on non-interactive zero knowledge proofs is proposed. The security of the proposal is presented by sequences of games without random oracles; furthermore, this scheme has a security proof for the property of privacy of the signer’s identity in comparison with the scheme proposed by Zhang et al. in 2007. In addition, this proposal compared to the scheme presented by Huang et al. in 2011 supports non-delegatability. The non-delegatability of our proposal is achieved since we do not use the common secret key shared between the signer and the designated verifier in our construction. Furthermore, if a signer delegates her signing capability which is derived from her secret key on a specific message to a third party, then, the third party cannot generate a valid designated verifier signature due to the relaxed special soundness of the non-interactive zero knowledge proof. To the best of our knowledge, this construction is the first attempt to generate a designated verifier signature scheme with non-delegatability in the standard model, while satisfying of non-delegatability property is loose.

This paper presents new explicit formulas for the point doubling, tripling and addition for Hessian curves and their equivalent Weierstrass curves over finite fields of characteristic three. The cost of basic point operations is lower than that of all previously proposed ones. The new doubling, mixed addition and tripling formulas in projective coordinates require 3M+2C, 8M+1C+1D and 4M+4C+1D respectively, where M, C and D is the cost of a field multiplication, a cubing and a multiplication by a constant. Finally, we present several examples of ordinary elliptic curves in characteristic three for high security levels.

Identity based encryption (IBE) has received great attention since Boneh and Franklin's breakthrough work on bilinear group based IBE [4]. Till now, many IBE schemes relying on bilinear groups with diff erent properties have been proposed [5, 25, 29, 14]. However, one part of the user's private key in all these IBE schemes is constructed as y = f(msk), where msk is the master key and y is an element in the underlying bilinear group G. In this paper, we propose a new IBE: one part of the private key is y = f(msk), where msk is the master key and y is an element in Z_p . Here p is the underlying bilinear group's prime order. By using some novel techniques, we prove this new IBE is semantic secure under the selective identity chosen plaintext attacks (IND-sID-CPA) in the standard model. Based on this IBE scheme, we construct an IND-ID-CCA secure identity based proxy re-encryption (IBPRE) scheme which is master secret secure and efficient for the proxy compared with other IND-ID-CCA (IBPRE) schemes.

Certificateless cryptography aims at combining the advantages of identity based and public key cryptography, so as to avoid the key escrow problem inherent in the identity based system and cumbersome certificate management in public key infrastructure. Signcryption achieves confidentiality and authentication simultaneously in an e

I propose the new key distribution system and attribute-based encryption scheme on non-commutative ring where the complexity required for enciphering and deciphering is small. As in this system encryption keys and decryption keys involve the attributes of each user, the system is adaptive for cloud computing systems. The security of this system is based on the complexity for solving the multivariate algebraic equations of high degree over finite field, that is, one of NP complete problems. So this system is immune from the Gröbner basis attacks. The key size of this system becomes to be small enough to handle.

We present in these papers a scheme, which bypasses the weakness presented in the existed scheme of IBE with random oracle. We propose, a secure scheme which project into Zp contrary to elliptic curve as with Boneh and Franklin. More, our scheme is basing in its study of simulation in the problem 4-EBDHP which is more efficient than q-BDHIP used by Skai Kasarah. We provide the prove of security of our scheme and we show its efficiency by comparison with the scheme declared above. Even if it we have a little cost in complexity, but as in the field cryptography we are more interested to the security, this makes our proposition more efficient.

In this paper, we present an identity-based encryption (IBE) scheme from lattices with efficient key revocation. We adopt multiple trapdoors from the Agrawal-Boneh-Boyen and Gentry-Peikerty-Vaikuntanathan lattice IBE schemes to realize key revocation, which in turn, makes use of binary-tree data structure. Using our scheme, key update requires logarithmic complexity in the maximal number of users and linear in the number of revoked users for the relevant key authority. We prove that our scheme is selective secure in the standard model and under the LWE assumption, which is as hard as the worst-case approximating short vectors on arbitrary lattices. Moreover, our key revocation techniques from lattices can be applied to obtain revocable functional encryption schemes in the similar setting.

Hierarchical inner product encryption (HIPE) and spatial encryption (SE) are two important classes of functional encryption (FE) that have a large number of applications. Although HIPE and SE both involve some notion of linear algebra, the former works in vectors while the latter is based on (affine) spaces. Moreover, they currently possess different properties in terms of security, anonymity (payload/attribute-hiding) and ciphertext sizes, for example. In this paper, we formally study the relation between HIPE and SE. In our work, we discover some interesting and novel property-preserving transformation techniques that enable generic construction of an SE scheme from an HIPE scheme, and vice versa.

Public-key encryption schemes rely for their IND-CPA security on per-message fresh randomness. In practice, randomness may be of poor quality for a variety of reasons, leading to failure of the schemes. Expecting the systems to improve is unrealistic. What we show in this paper is that we can, instead, improve the cryptography to offset the lack of possible randomness. We provide public-key encryption schemes that achieve IND-CPA security when the randomness they use is of high quality, but, when the latter is not the case, rather than breaking completely, they achieve a weaker but still useful notion of security that we call IND-CDA. This hedged public-key encryption provides the best possible security guarantees in the face of bad randomness. We provide simple RO-based ways to make in-practice IND-CPA schemes hedge secure with minimal software changes. We also provide non-RO model schemes relying on lossy trapdoor functions (LTDFs) and techniques from deterministic encryption. They achieve adaptive security by establishing and exploiting the anonymity of LTDFs which we believe is of independent interest.

We propose a fully private fingerprint matching protocol that compares two fingerprints based on the most widely-used minutia-based fingerprint matching algorithm. The protocol enables two parties, each holding a private fingerprint, to find out if their fingerprints belong to the same individual. Unlike previous works, we do not make any simplifying assumption on the matching algorithm or use generic multiparty computation protocols in our constructions. We employ a commonly-used algorithm that works by first comparing minutia pairs from the two fingerprints based on their types, locations, and orientations, and then checking if the number of matching minutia pairs is more than a threshold, and we propose a concrete, scalable, and modular protocol. We prove security against honest-but-curious adversaries and discuss how security against malicious adversaries can be achieved using standard cryptographic techniques. Our protocol is realized using common cryptographic primitives and do not require pairing- or lattice-based cryptography.

Verifiable random functions (VRF) and selectively-convertible undeniable signature (SCUS) schemes were proposed independently in the literature. In this paper, we observe that they are tightly related. This directly yields several deterministic SCUS schemes based on existing VRF constructions. In addition, we create a new probabilistic SCUS scheme, which is very compact. The confirmation and disavowal protocols of these SCUS are efficient, and can be run either sequentially, concurrently, or arbitrarily. These protocols are based on what we call zero-knowledge protocols for generalized DDH and non-DDH, which are of independent interest.

While it is generally believed that due to their large public key sizes code based public key schemes cannot be conveniently used when memory-constrained devices are involved, we propose an approach for Public Key Infrastructure (PKI) scenarios which totally eliminates the need to store public keys of communication partners. Instead, all the necessary computation steps are performed during the transmission of the key. We show the feasibility of the approach through an example implementation and give arguments that it will be possible for a smart card controller to carry out the associated computations to sustain the transmission rates of possible future high speed contactless interfaces.

We introduce a natural cryptographic functionality called \emph{functional re-encryption}. Informally, this functionality, for a public-key encryption scheme and a function $F$ with $n$ possible outputs, transforms (``re-encrypts") an encryption of a message $m$ under an ``input public key" $\pk$ into an encryption of the same message $m$ under one of the $n$ ``output public keys", namely the public key indexed by $F(m)$. In many settings, one might require that the program implementing the functional re-encryption functionality should reveal nothing about both the input secret key $\sk$ as well as the function $F$. As an example, consider a user Alice who wants her email server to share her incoming mail with one of a set of $n$ recipients according to an access policy specified by her function $F$, but who wants to keep this access policy private from the server. Furthermore, in this setting, we would ideally obtain an even stronger guarantee: that this information remains hidden even when some of the $n$ recipients may be corrupted. To formalize these issues, we introduce the notion of \emph{collusion-resistant obfuscation} and define this notion with respect to average-case secure obfuscation (Hohenberger \emph{et al.} - TCC 2007). We then provide a construction of a functional re-encryption scheme for any function $F$ with a polynomial-size domain and show that it satisfies this notion of collusion-resistant obfuscation. We note that collusion-resistant security can be viewed as a special case of dependent auxiliary input security (a setting where virtually no positive results are known), and this notion may be of independent interest. Finally, we show that collusion-resistant obfuscation of functional re-encryption for a function $F$ gives a way to obfuscate $F$ in the sense of Barak {\em et al.} (CRYPTO 2001), indicating that this task is impossible for arbitrary (polynomial-time computable) functions $F$.

Signcryption schemes provide an efficient messaging system for data that needs to be sent with data confidentiality, data integrity and data origin authentication. However, the bandwidth overhead for the use of signcryption in a network in which a large number of messages need to be sent may be high. Motivated by aggregate signature schemes, we propose the concept of an aggregate signcryption scheme. An aggregate signcryption scheme allows distinct signcryption ciphertexts intended for the same recipient to be merged into a single signcryption ciphertext of smaller size without losing any of their security guarantees. This has the potential to provide significant bandwidth savings. We propose security models for this scheme, analyse the trivial generic constructions, propose an efficient new scheme, and analyse the bandwidth requirements of these schemes for a practical distributed database application.

In \cite{SS11}, Sakumoto et al. presented a new multivariate identification scheme, whose security is based solely on the MQ-Problem of solving systems of quadratic equations over finite fields. In this paper we extend this scheme to a threshold ring identification and signature scheme. Our scheme is the first multivariate scheme of this type and generally the first multivariate signature scheme with special properties. Despite the fact that we need more rounds to achieve given levels of security, the signatures are at least twice shorter than those obtained by other post-quantum (e.g. code based) constructions. Furthermore, our scheme offers provable security, which is quite a rare fact in multivariate cryptography.

In this paper, we present several efficient fault attacks against implementations of RSA-CRT signatures that use modular exponentiation algorithms based on Montgomery multiplication. They apply to any padding function, including randomized paddings, and as such are the first fault attacks effective against RSA-PSS. The new attacks work provided that a small register can be forced to either zero, or a constant value, or a value with zero high-order bits. We show that these models are quite realistic, as such faults can be achieved against many proposed hardware designs for RSA signatures.

We show that homomorphic evaluation of (wide enough) arithmetic circuits can be accomplished with only polylogarithmic overhead. Namely, we present a construction of fully homomorphic encryption (FHE) schemes that for security parameter $\secparam$ can evaluate any width-$\Omega(\secparam)$ circuit with $t$ gates in time $t\cdot polylog(\secparam)$. To get low overhead, we use the recent batch homomorphic evaluation techniques of Smart-Vercauteren and Brakerski-Gentry-Vaikuntanathan, who showed that homomorphic operations can be applied to "packed" ciphertexts that encrypt vectors of plaintext elements. In this work, we introduce permuting/routing techniques to move plaintext elements across these vectors efficiently. Hence, we are able to implement general arithmetic circuit in a batched fashion without ever needing to "unpack" the plaintext vectors. We also introduce some other optimizations that can speed up homomorphic evaluation in certain cases. For example, we show how to use the Frobenius map to raise plaintext elements to powers of~$p$ at the "cost" of a linear operation.

Aggregate signatures are useful in special areas where the signatures on many different messages generated by many different users need to be compressed. Recently, Xiong et al. proposed a certificateless aggregate signature scheme provably secure in the random oracle model under the Computational Diffie-Hellman assumption. Unfortunately, by giving concrete attacks, we indicate that Xiong et al. aggregate signature scheme does not meet the basic requirement of unforgeability.

Bogdanov and Lee suggested a homomorphic public-key encryption scheme based on error correcting codes. The underlying public code is a modified Reed-Solomon code obtained from inserting a zero submatrix in the Vandermonde generating matrix defining it. The columns that define this submatrix are kept secret and form a set $L$. We give here a distinguisher that detects if one or several columns belong to $L$ or not. This distinguisher is obtained by considering the code generated by component-wise products of codewords of the public code (the so called ``square code''). This operation is applied to punctured versions of this square code obtained by picking a subset $I$ of the whole set of columns. It turns out that the dimension of the punctured square code is directly related to the cardinality of the intersection of $I$ with $L$. This allows an attack which recovers the full set $L$ and which can then decrypt any ciphertext.

This paper presents an attack in the multi-user setting on various public-key encryption schemes standardized in IEEE 1363a, SECG SEC 1 and ISO 18033-2. The multi-user setting is a security model proposed by Bellare et al., which allows adversaries to simultaneously attack multiple ciphertexts created by one or more users. An attack is considered successful if the attacker learns information about any of the plaintexts. We show that many standardized public-key encryption schemes are vulnerable in this model, and give ways to prevent the attack. We also show that the key derivation function and pseudorandom generator used to implement a hybrid encryption scheme must be secure in the multi-user setting, in order for the overall primitive to be secure in the multi-user setting. As an illustration of the former, we show that using HKDF (as standardized in NIST SP 800-56C) as a key derivation function for certain standardized hybrid public-key encryption schemes is insecure in the multi-user setting.

This paper presents a heuristic attack on the fully homomorphic encryption over the integers by using lattice reduction algorithm. Our result shows that the FHE in [DGHV10] is not secure for some parameter settings. We also present an improvement scheme to avoid the lattice attack in this paper.

A group encryption scheme allows anyone to form a ciphertext for a given group member while keeping the receiver's identity private. At the same time, the encryptor is capable of proving that some (anonymous) group member is able to decrypt the ciphertext and, optionally, that the corresponding plaintext satisfies some \apriori\ relation (to prevent sending bogus messages). Finally, in case of a dispute, the identity of the intended receiver can be recovered by a designated authority. In this paper, we abstract a generic approach to construct group encryption schemes. We also introduce several new implementation tricks. As a result, we obtain group encryption schemes that significantly improve the state of the art. Both interactive and non-interactive constructions are considered.

It is well-known that the k-wise product of one-way functions remains one-way, but may no longer be when the k inputs are correlated. At TCC 2009, Rosen and Segev introduced a new notion known as Correlated Product secure functions. These functions have the property that a k-wise product of them remains one-way even under correlated inputs. Rosen and Segev gave a construction of injective trapdoor functions which were correlated product secure from the existence of Lossy Trapdoor Functions (introduced by Peikert and Waters in STOC 2008). The first main result of this work shows the surprising fact that a family of correlated product secure functions can be constructed from any one-way function. Because correlated product secure functions are trivially one-way, this shows an equivalence between the existence of these two cryptographic primitives. In the second main result of this work, we consider a natural decisional variant of correlated product security. Roughly, a family of functions are Decisional Correlated Product (DCP) secure if $f_1(x_1),\ldots,f_k(x_1)$ is indistinguishable from $f_1(x_1),\ldots,f_k(x_k)$ when $x_1,\ldots,x_k$ are chosen uniformly at random. We argue that the notion of Decisional Correlated Product security is a very natural one. To this end, we show a parallel from the Discrete Log Problem and Decision Diffie-Hellman Problem to Correlated Product security and its decisional variant. This intuition gives very simple constructions of PRGs and IND-CPA encryption from DCP secure functions. Furthermore, we strengthen our first result by showing that the existence of DCP secure one-way functions is also equivalent to the existence of any one-way function. When considering DCP secure functions with trapdoors, we give a construction based on Lossy Trapdoor Functions, and show that any DCP secure function family with trapdoor satisfy the security requirements for Deterministic Encryption as defined by Bellare, Boldyreva and O'Neill in CRYPTO 2007. In fact, we also show that definitionally, DCP secure functions with trapdoors are a strict subset of Deterministic Encryption functions by showing an example of a Deterministic Encryption function which according to the definition is not a DCP secure function.

A new multivariate public-key cryptosystem (MPKC) with the security based on the difficulty of the prime factoring is proposed. Unlike conventional cryptosystems such as RSA, most MPKCs are expected secure against quantum computers, and their operation of encryption and decryption is expected quick, because they do not need exponential operation. However, their security against quantum computers is very difficult to prove mathematically. We propose a new MPKC based on sequential solution method, assuming the security against von Neumann computers, whose attack seems as difficult as prime factoring. This cryptosystem is applicable to both encryption and signature.

In this paper we port Brakerski's fully homomorphic scheme based on the Learning With Errors (LWE) problem to the ring-LWE setting. We introduce two optimised versions of relinearisation that not only result in a smaller relinearisation key, but also faster computations. We provide a detailed, but simple analysis of the various homomorphic operations, such as multiplication, relinearisation and bootstrapping, and derive tight worst case bounds on the noise caused by these operations. The analysis of the bootstrapping step is greatly simplified by using a modulus switching trick. Finally, we derive concrete parameters for which the scheme provides a given level of security and becomes fully homomorphic.

In the attribute-based encryption (ABE) model, attributes (as opposed to identities) are used to encrypt messages, and all the receivers with qualifying attributes can decrypt the ciphertext. However, compromised attribute keys may affect the communications of many users who share the same access control policies. We present the notion of forward-secure attribute-based encryption (fs-ABE) and give a concrete construction based on bilinear map and decisional bilinear Diffie-Hellman assumption. Forward security means that a compromised private key by an adversary at time $t$ does not break the confidentiality of the communication that took place prior to $t$. We describe how to achieve both forward security and encryption with attributes, and formally prove our security against the adaptive chosen-ciphertext adversaries. Our scheme is non-trivial, and the key size only grows polynomially with $\log N$ (where $N$ is the number of time periods). We further generalize our scheme to support the individualized key-updating schedule for each attribute, which provides a finer granularity for key management. Our insights on the required properties that an ABE scheme needs to possess in order to be forward-secure compatible are useful beyond the specific fs-ABE construction given. We raise an open question at the end of the paper on the escrow problem of the master key in ABE schemes.

Network Coding is a routing technique where each node may actively modify the received packets before transmitting them. While this departure from passive networks improves throughput and resilience to packet loss it renders transmission susceptible to {\em pollution attacks} where nodes can misbehave and change in a malicious way the messages transmitted. Nodes cannot use standard signature schemes to authenticate the modified packets: this would require knowledge of the original sender's signing key. Network coding signature schemes offer a cryptographic solution to this problem. Very roughly, such signatures allow signing vector spaces (or rather bases of such spaces). Furthermore, these signatures are homomorphic: given signatures on a set of vectors it is possible to create signatures for any linear combination of these vectors. Designing such schemes is a difficult task, and the few existent constructions either rely on random oracles or are rather inefficient. In this paper we introduce two new network coding signature schemes. Both of our schemes are provably secure in the standard model, rely on standard assumptions, {\em and} are in the same efficiency class with previous solutions based on random oracles.

We propose a general framework that converts (ordinary) signature schemes having certain properties into linearly homomorphic signature schemes, i.e., schemes that allow authentication of linear functions on signed data. The security of the homomorphic scheme follows from the same computational assumption as is used to prove security of the underlying signature scheme. We show that the following signature schemes have the required properties and thus give rise to secure homomorphic signatures in the standard model: - The scheme of Waters (Eurocrypt 2005), secure under the computational Diffie-Hellman asumption in bilinear groups. - The scheme of Boneh and Boyen (Eurocrypt 2004, J. Cryptology 2008), secure under the $q$-strong Diffie-Hellman assumption in bilinear groups. - The scheme of Gennaro, Halevi, and Rabin (Eurocrypt 1999), secure under the strong RSA assumption. - The scheme of Hohenberger and Waters (Crypto 2009), secure under the RSA assumption. Our systems not only allow weaker security assumptions than were previously available for homomorphic signatures in the standard model, but also are secure in a model that allows a stronger adversary than in other proposed schemes. Our framework also leads to efficient linearly homomorphic signatures that are secure against our stronger adversary under weak assumptions (CDH or RSA) in the random oracle model; all previous proofs of security in the random oracle model break down completely when faced with our stronger adversary.

We introduce and study a new type of DDH-like assumptions based on groups of prime order q. Whereas standard DDH is based on encoding elements of F_{q} ``in the exponent'' of elements in the group, we ask what happens if instead we put in the exponent elements of the extension ring R_f= \F_{q}[X]/(f) where f can be any degree-d polynomial. We show that solving the decision problem that follows naturally reduces to the case where f is irreducible. This variant is called the d-DDH problem, where 1-DDH is standard DDH. Essentially any known cryptographic construction based on DDH can be immediately generalized to use instead d-DDH, and we show in the generic group model that d-DDH is harder than DDH. This means that virtually any application of DDH can now be realized with the same (amortized) efficiency, but under a potentially weaker assumption. On the negative side, we also show that d-DDH, just like DDH, is easy in bilinear groups. This motivates our suggestion of a different type of assumption, the d-vector DDH problems (VDDH), which are based on f(X)= X^d, but with a twist to avoid the problems with reducible polynomials. We show in the generic group model that VDDH is hard in bilinear groups and that in fact the problems become harder with increasing d and hence form an infinite hierarchy. We show that hardness of VDDH implies CCA-secure encryption, efficient Naor-Reingold style pseudorandom functions, and auxiliary input secure encryption, a strong form of leakage resilience. This can be seen as an alternative to the known family of k-linear assumptions.

Tate pairing computation consists of two parts: Miller step and final exponentiation step. In this paper, we investigate how to accelerate the final exponentiation step. Consider an order $r$ subgroup of an elliptic curve defined over $\Fq$ with embedding degree $k$. The final exponentiation in the Tate pairing is an exponentiation of an element in $\Fqk$ by $(q^k-1)/r$. The hardest part of this computation is to raise to the power $\lam:=\varphi_k(q)/r$. Write it as $\lam=\lam_0+\lam_1q+\cdots+\lam_{d-1}q^{d-1}$ in the $q$-ary representation. When using multi-exponentiation techniques with precomputation, the final exponentiation cost mostly depends on $\kappa(\lambda)$, the size of the maximum of $|\lambda_i|$. In many parametrized pairing-friendly curves, the value $\kappa$ is about $\left(1-\frac{1}{\rho\varphi(k)}\right)\log q$ where $\rho=\log q/\log r$, while random curves will have $\kappa \approx \log q$. We analyze how this small $\kappa$ is obtained for parametrized elliptic curves, and show that $\left(1-\frac{1}{\rho\varphi(k)}\right)\log q$ is almost optimal in the sense that for all known construction methods of parametrized pairing-friendly curves it is the lower bound. This method is useful, but has a limitation that it can only be applied to only parametrized curves and excludes many of elliptic curves. In the second part of our paper, we propose a method to obtain a modified Tate pairing with smaller $\kappa$ for {\em any elliptic curves}. More precisely, our method finds an integer $m$ such that $\kappa(m\lambda)=\left(1-\frac{1}{\rho\varphi(k)}\right)\log q$ efficiently using lattice reduction. Using this modified Tate pairing, we can reduce the number of squarings in the final exponentiation by about $\left(1-\frac{1}{\rho\varphi(k)}\right)$ times from the usual Tate pairing. We apply our method to several known pairing friendly curves to verify the expected speedup.

Several schemes have been proposed towards the fast encryption and decryption in RSA and its variants. One popular idea is to use integers having low Hamming weight in the preparation of the decryption exponents. This is to reduce the multiplication effort in the square and multiply method in the exponentiation routine, both in encryption and decryption. In this paper we show that such schemes are insecure in CRT-RSA when the encryption exponent is small (e.g., $e = 2^{16}+1$). In particular, we show that the CRT-RSA schemes presented in SAC 1996 and ACISP 2005 with low weight decryption exponents can be broken in a few minutes in certain cases. Further, the scheme of CT-RSA 2010, where the decryption exponents are not of low weight but they have large low weight factors, can also be cryptanalysed. To mount the attack, we exploit the heuristic proposed by Henecka et al (Crypto 2010) that is capable of correcting errors in the secret parameters when the encryption exponent is small. In the process, we identify a few modifications of the error correction strategy that provides significantly improved experimental outcome and also beats the theoretical bounds given in the work of Henecka et al.

Certificateless public key encryption (CLE) and certificate based encryption (CBE) are two novel public key cryptographic primitives requiring no authenticity verification of the recipient's public key. Both of them are motivated to simultaneously solve the heavy certificate management problem inherent in the traditional public key encryption (PKE) and the key escrow problem inherent in the identity-based encryption (IBE). It is an attractive cryptographic task to formally explore the relation between CBE and CLE. In 2005, Al-Riyami and Paterson proposed one general conversion from CLE to CBE. Shortly later, Kang and Park pointed out a flaw in the security proof of Al-Riyami-Paterson conversion. In 2012, Wu et al. proposed another generic conversion from CLE to CBE. Compared with Al-Riyami-Paterson conversion, Wu et al.'s method can be proved secure, but it has to additionally involve collision resistant hash functions. It remains an open problem whether the generic conversion due to Al-Riyami and Paterson, which is very neat, is provably secure. We aim to solve this open problem. First, we formalize CLE's new security model, featured by introducing a new security property overlooked by previous security models. With this new security model as the basic technique, we succeed in proving that the Al-Riyami-Paterson generic conversion from CLE to CBE is secure, if the CLE scheme is secure in our new security model. A concrete provably secure CBE scheme is presented to demonstrate the application of our result.

Taking advantage of a technique that allows to safely distribute a private key among decryption servers we introduce a new identity-based threshold scheme, proven secure in the random oracle model. This new paring-based scheme features a lot of improvements compared to other schemes that can be found in the literature. Among them the two most noticeable ones are, the efficiency, by reducing the number of pairing computations, and the ability for a user to generate and share a private key without requiring any access to a PKG.

In this paper we present a new algorithm to attack lattice based cryptosystems by solving a problem over real numbers. In the case of the NTRU cryptosystem, if we assume the additional information on the modular operations, we can break the NTRU cryptosystems completely by getting the secret key. We believe that this fact was not known before.

In this paper, we present a new class of public-key cryptosystems, K(XV)SE(2)PKC realizing the coding rate of exactly 1.0, based on Reed-Solomon codes(RS codes). We show that K(XV)SE(2)PKC is secure against the various attacks including the attacks based on the Gröbner basis calcula&#65364;ion (Gröbner basis attack, GB attack) and a linear transformation attack.

We propose a new identity-based threshold signature (IBTHS) scheme from bilinear pairings enjoying the following advantages in efficiency, security and functionality. The round-complexity of the threshold signing protocol is optimal since each party pays no other communication cost except broadcasting one single message. The computational complexity of the threshold signing procedure is considerably low since there appears no other time-consuming pairing except two pairings for verifying each signature shares. The communication channel requirement of the threshold signing procedure is the lowest since the broadcast channel among signers is enough. It is proved secure with optimal resilience in the standard model. It is the private key associated with an identity rather than a master key of the Public Key Generator (PKG) that is shared among signature generation servers. All these excellent properties are due to our new basic technique by which the private key in the bilinear group is indirectly shared through simply sharing an element in the finite field.

We performed a sanity check of public keys collected on the web. Our main goal was to test the validity of the assumption that different random choices are made each time keys are generated. We found that the vast majority of public keys work as intended. A more disconcerting finding is that two out of every one thousand RSA moduli that we collected offer no security. Our conclusion is that the validity of the assumption is questionable and that generating keys in the real world for ``multiple-secrets'' cryptosystems such as RSA is significantly riskier than for ``single-secret'' ones such as ElGamal or (EC)DSA which are based on Diffie-Hellman.

In this study, a novel strong designated verifier signature scheme based on bilinear pairings with provable security in the standard model is proposed, while the existing ones are secure in the random oracle model. In 2007 and 2011, two strong designated verifier signature schemes in the standard model are proposed by Huang et al. and Zhang et al., respectively; in the former, the property of privacy of the signer’s identity is not proved and the security of the latter is based on the security of a pseudorandom function. Our proposal can deal with the aforementioned drawbacks of the previous schemes. Furthermore, it satisfies non-delegatability for signature verification

Identity-based encryption (IBE) is an exciting alternative to public-key encryption, as IBE eliminates the need for a Public Key Infrastructure (PKI). Any setting, PKI- or identity-based, must provide a means to revoke users from the system. Efficient revocation is a well-studied problem in the traditional PKI setting. However in the setting of IBE, there has been little work on studying the revocation mechanisms. The most practical solution requires the senders to also use time periods when encrypting, and all the receivers (regardless of whether their keys have been compromised or not) to update their private keys regularly by contacting the trusted authority. We note that this solution does not scale well -- as the number of users increases, the work on key updates becomes a bottleneck. We propose an IBE scheme that significantly improves key-update efficiency on the side of the trusted party (from linear to logarithmic in the number of users), while staying efficient for the users. Our scheme builds on the ideas of the Fuzzy IBE primitive and binary tree data structure, and is provably secure.

Motivated by applications in large storage systems, we initiate the study of incremental deterministic public-key encryption. Deterministic public-key encryption, introduced by Bellare, Boldyreva, and O'Neill (CRYPTO '07), provides a realistic alternative to randomized public-key encryption in various scenarios where the latter exhibits inherent drawbacks. A deterministic encryption algorithm, however, cannot satisfy any meaningful notion of security for low-entropy plaintexts distributions, and Bellare et al. demonstrated that a strong notion of security can in fact be realized for relatively high-entropy plaintext distributions. In order to achieve a meaningful level of security, a deterministic encryption algorithm should be typically used for encrypting rather long plaintexts for ensuring a sufficient amount of entropy. This requirement may be at odds with efficiency constraints, such as communication complexity and computation complexity in the presence of small updates. Thus, a highly desirable property of deterministic encryption algorithms is incrementality: small changes in the plaintext translate into small changes in the corresponding ciphertext. We present a framework for modeling the incrementality of deterministic public-key encryption. Within our framework we propose two schemes, which we prove to enjoy an optimal tradeoff between their security and incrementality up to small polylogarithmic factors. Our first scheme is a generic method which can be based on any deterministic public-key encryption scheme, and in particular, can be instantiated with any semantically-secure (randomized) public-key encryption scheme in the random oracle model. Our second scheme is based on the Decisional Diffie-Hellman assumption in the standard model. The approach underpinning our schemes is inspired by the fundamental ``sample-then-extract'' technique due to Nisan and Zuckerman (JCSS '96) and refined by Vadhan (J. Cryptology '04), and by the closely related notion of ``locally-computable extractors'' due to Vadhan. Most notably, whereas Vadhan used such extractors to construct private-key encryption schemes in the bounded-storage model, we show that techniques along these lines can also be used to construct incremental public-key encryption schemes.

This paper proposes the first inner product encryption (IPE) scheme that is adaptively secure and fully attribute-hiding (attribute-hiding in the sense of the definition by Katz, Sahai and Waters), while the existing IPE schemes are either fully attribute-hiding but selectively secure or adaptively secure but weakly attribute-hiding. The proposed IPE scheme is proven to be adaptively secure and fully attribute-hiding under the decisional linear assumption in the standard model. The IPE scheme is comparably as efficient as the existing attribute-hiding IPE schemes. We also present a variant of the proposed IPE scheme with the same security that achieves shorter public and secret keys. A hierarchical IPE scheme can be constructed that is also adaptively secure and fully attribute-hiding under the same assumption. In this paper, we extend the dual system encryption technique by Waters into a more general manner, in which new forms of ciphertext and secret keys are employed and new types of information theoretical tricks are introduced along with several forms of computational reduction.

The Schnorr signature scheme has been known to be provably secure in the Random Oracle Model under the Discrete Logarithm (DL) assumption since the work of Pointcheval and Stern (EUROCRYPT '96), at the price of a very loose reduction though: if there is a forger making at most $q_h$ random oracle queries, and forging signatures with probability $\varepsilon_F$, then the Forking Lemma tells that one can compute discrete logarithms with constant probability by rewinding the forger $\mathcal{O}(q_h/\varepsilon_F)$ times. In other words, the security reduction loses a factor $\mathcal{O}(q_h)$ in its time-to-success ratio. This is rather unsatisfactory since $q_h$ may be quite large. Yet Paillier and Vergnaud (ASIACRYPT 2005) later showed that under the One More Discrete Logarithm (OMDL) assumption, any \emph{algebraic} reduction must lose a factor at least $q_h^{1/2}$ in its time-to-success ratio. This was later improved by Garg~\emph{et al.} (CRYPTO 2008) to a factor $q_h^{2/3}$. Up to now, the gap between $q_h^{2/3}$ and $q_h$ remained open. In this paper, we show that the security proof using the Forking Lemma is essentially the best possible. Namely, under the OMDL assumption, any algebraic reduction must lose a factor $f(\varepsilon_F)q_h$ in its time-to-success ratio, where $f\le 1$ is a function that remains close to 1 as long as $\varepsilon_F$ is noticeably smaller than 1. Using a formulation in terms of expected-time and queries algorithms, we obtain an optimal loss factor $\Omega(q_h)$, independently of $\varepsilon_F$. These results apply to other signature schemes based on one-way group homomorphisms, such as the Guillou-Quisquater signature scheme.

We describe a compression technique that reduces the public key size of van Dijk, Gentry, Halevi and Vaikuntanathan's (DGHV) fully homomorphic scheme over the integers from \lambda^7 to \lambda^5. Our variant remains semantically secure, but in the random oracle model. We obtain an implementation of the full scheme with a 10.1 MB public key instead of 802 MB using similar parameters as in \cite{cmnt2011}. Additionally we show how to extend the quadratic encryption technique of \cite{cmnt2011} to higher degrees, to obtain a shorter public-key for the basic scheme. This paper also describes a new modulus switching technique for the DGHV scheme that enables to use the new FHE framework without bootstrapping from Brakerski, Gentry and Vaikuntanathan with the DGHV scheme. Finally we describe an improved attack against the Approximate GCD Problem on which the DGHV scheme is based, with complexity 2^\rho instead of 2^{3\rho/2}.

Waters, in 2009, introduced an important technique, called dual-system encryption, to construct identity-based encryption (IBE) and related schemes. The resulting IBE scheme was described in the setting of symmetric pairing. A key feature of the construction is the presence of random tags in the ciphertext and decryption key. Later work by Lewko and Waters has removed the tags and proceeding through composite-order pairings has led to a more efficient dual-system IBE scheme using asymmetric pairings whose security is based on non-standard but static assumptions. In this work, we have systematically simplified Waters 2009 IBE scheme in the setting of asymmetric pairing. The simplifications retain tags used in the original description. This leads to several variants, the first one of which is based on standard assumptions and in comparison to Waters original scheme reduces ciphertexts and keys by two elements each. Going through several stages of simplifications, we finally obtain a simple scheme whose security can be based on two standard assumptions and a natural and minimal extension of the decision Diffie-Hellman problem for asymmetric pairing groups. The scheme itself is also minimal in the sense that apart from the tags, both encryption and key generation use exactly one randomiser each. This final scheme is more efficient than both the previous dual-system IBE scheme in the asymmetric setting due to Lewko and Waters and the more recent dual-system IBE scheme due to Lewko. We extend the IBE scheme to hierarchical IBE (HIBE) and broadcast encryption (BE) schemes. Both primitives are secure in their respective full models and have better efficiencies compared to previously known schemes offering the same level and type of security.

Blind signatures provide a mechanism for achieving privacy and anonymity whereby a user gets the signer to sign a message of his choice without the signer learning the content of the message, nor linking message/signature request pairs when he sees the nal signature. In this paper, we construct a blind signature that requires minimal interaction (two moves) between the user and the signer, and which results in a signature which is a signature with respect to a standard (i.e. non-blind) signature scheme. The signature request protocol is akin to the classic, blind-unblind methodology used for RSA blind signatures in the random oracle model; whilst the output signature is a standard Camenisch-Lysyanskaya signature in bilinear groups. The scheme is secure in the common reference string model, assuming a discrete logarithm related assumption in bilinear groups; namely a new variant of the LRSW assumption. We provide evidence for the hardness of our new variant of the LRSW by showing it is intractable in the generic group model.

Digital signature is one of the basic primitives in cryptography. A common paradigm of obtaining signatures, known as the Fiat-Shamir (FS) paradigm, is to collapse any &#931;-protocol (which is 3-round public-coin honest-verifier zero-knowledge) into a non-interactive scheme with hash functions that are modeled to be random oracles (RO). The Digital Signature Standard (DSS) and Schnorr’s signature schemes are two salient examples following the FS-paradigm. In this work, we present a modified Fiat-Shamir paradigm, named challenge-divided Fiat-Shamir paradigm, which is applicable to a variant of &#931;-protocol with divided random challenges. This new paradigm yields a new family of (online/offline efficient) digital signatures from challenge-divided &#931;-protocols, including in particular a variant of Schnorr’s signature scheme called challenge-divided Schnorr signature. We then present a formal analysis of the challenge-divided Schnorr signature in the random oracle model. Finally, we give comparisons between the challenge-divided Schnorr signature and DSS and Schnorr’s signature, showing that the newly developed challenge-divided Schnorr signature can enjoy better (online/offline) efficiency (besides provable security in the random oracle model). Of independent interest is a new forking lemma, referred to as divided forking lemma, for dealing with multiple ordered rewinding points in the RO model, which is of independent interest and can be applied to analyzing other cryptographic schemes in the RO model.

Server-aided verification(SAV) signature is useful for power-constrained devices since a powerful server can assist in performing costly operations such as pairing operations. Wu et al. [13] defined three security notions for SAV protocol to prevent a server from convincing a verifier that an invalid signature is valid. Security against strong collusion attack provides the strongest security guarantee among these notions. They [13] constructed SAV protocols that meet the requirement of these notions respectively. But they did not provide concrete running time to show that the running time of a verifier in their SAV protocol is strictly less than that of a verifier in the original verification protocol. In addition, a problem left open by their work is to design SAV signature which is unforgeable without random oracles as well as sound against strong collusion attack. To address the above issues, we first choose to design a SAV protocol called SAV-Hofheinz for a short signature proposed by Hofheinz unforgeable in the standard model. Then we implement SAV-Hofheinz by the PBC library and shows that the running time of a verifier in SAV-Hofheinz is strictly less than that of a verifier in the verification protocol of Hofheinz short signature.

We complete the study of [23] and [27] about Miller's algorithm. Miller's algorithm is a central step to compute the Weil, Tate and Ate pairings. The aim of this article is to analyze the weakness of Miller's algorithm when it undergoes a fault attack. We prove that Miller's algorithm is vulnerable to a fault attack which is valid in all coordinate systems, through the resolution of a nonlinear system. We highlight the fact that putting the secret as the rst argument of the pairing is not a countermeasure. This article is an extensed version of the article [15].

In tandem with recent progress on computing on encrypted data via fully homomorphic encryption, we present a framework for computing on *authenticated* data via the notion of slightly homomorphic signatures, or P-homomorphic signatures. With such signatures, it is possible for a third party to derive a signature on the object m' from a signature of m as long as P(m,m')=1 for some predicate P which captures the ``authenticatable relationship" between m' and m. Moreover, a derived signature on m' reveals no extra information about the parent m. Our definition is carefully formulated to provide one unified framework for a variety of distinct concepts in this area, including arithmetic, homomorphic, quotable, redactable, transitive signatures and more. It includes being unable to distinguish a derived signature from a fresh one even when given the original signature. The inability to link derived signatures to their original sources prevents some practical privacy and linking attacks, which is a challenge not satisfied by most prior works. Under this strong definition, we then provide generic constructions for all univariate and closed predicates, and specific efficient constructions for a broad class of natural predicates such as quoting, subsets, weighted sums, averages, and Fourier transforms. To our knowledge, these are the first efficient constructions for these predicates (excluding subsets) that provably satisfy this strong security notion.