The total computation of the generation and verification of personnel identification or digital signature is heavy. For many schemes of them, the total computation is not less than hundreds of modular multiplications. Efficient schemes of personnel identification and digital signature were proposed, which require no more than 10 modular multiplications on generation and verification of challenge-response or digital signature. However, the schemes are weak in security. The paper will show that by interception of a transcript of communications between the prover and verifier, the private key of the prover is revealed.

We present a new key exchange primitive based on the decomposition problem over non-commutative groups. Different from the key establishment schemes that rely on the decomposition problem where the problem is decomposing an element into three parts where the middle piece is known, our scheme relies on decomposing an element into three parts, all unknown. We call this problem "Triple Decomposition Problem". This seems to be a harder problem because it requires quadratic systems to be solved instead of linear systems. We discuss the new primitive over two different protocols. The underlying problems in the two protocols differ slightly. We discuss the system and the underlying problems in one of the protocols in detail over braid groups. We manage to provide a setting which resists against linear algebra attacks and length based attacks.

This paper demonstrates that a certificateless signature scheme recently proposed by Gorantla and Saxena is insecure. It is shown that an adversary who replaces the public key of a signer can then forge valid signatures for that signer without knowledge of the signer's private key.

The notion of concurrent signatures was introduced by Chen, Kudla and Paterson in their seminal paper in Eurocrypt 2004. In concurrent signature schemes, two entities can produce two signatures that are not binding, until an extra piece of information (namely the keystone) is released by one of the parties. Upon release of the keystone, both signatures become binding to their true signers concurrently. In ICICS 2005, two identity-based perfect concurrent signature schemes were proposed by Chow and Susilo. In this paper, we show that these two schemes are unfair, in which the initial signer can cheat the matching signer. We present a formal definition of ID-based concurrent signatures which redress the flaw of Chow et al.'s definition and then propose two simple but significant improvements to fix our attacks.

The notion of concurrent signatures was recently introduced by Chen, Kudla and Paterson. In concurrent signature schemes, two entities can produce two signatures that are not binding, until an extra piece of information (namely the keystone) is released by one of the parties. Subsequently, it was noted that the concurrent signature scheme proposed in the seminal paper cannot provide perfect ambiguity. Then, the notion of perfect concurrent signatures was introduced. In this paper, we define the notion of identity-based (or ID-based) perfect concurrent signature schemes. We provide the first generic construction of (ID-based) perfect concurrent signature schemes from ring signature schemes. Using the proposed framework, we give two concrete ID-based perfect concurrent signature schemes based on two major paradigms of ID-based ring signature schemes. Security proofs are based on the random oracle model.

The paper cryptanalyses a public-key cryptosystem recently proposed by Grigoriev and Ponomarenko, which encrypts an element from a fixed finite group defined in terms of generators and relations to produce a ciphertext from SL(2, Z). The paper presents a heuristic method for recovering the secret key from the public key, and so this cryptosystem should not be used in practice.

Abstract. An encrypted email is sent from Bob to Alice. A gateway wants to check whether a certain keyword exists in an email or not for some reason (e.g. routing). Nevertheless Alice does not want the email to be decrypted by anyone except her including the gateway itself. This is a scenario where public key encryption with keyword search (PEKS) is needed. In this paper we construct a new scheme (KR-PEKS) the KResilient Public Key Encryption with Keyword Search. The new scheme is secure under a chosen keyword attack without the random oracle. The ability of constructing a Public Key Encryption with Keyword Search from an Identity Based Encryption was used in the construction of the KR-PEKS. The security of the new scheme was proved by showing that the used IBE has a notion of key privacy. The scheme was then modified in two different ways in order to fulfill each of the following: the first modification was done to enable multiple keyword search and the other was done to remove the need of secure channels.

Rewinding techniques form the essence of many security reductions including proofs for identification and signature schemes. We propose a simple and modular approach for the construction of such proofs. Straightforward applications of our central result include, but are not limited to, the security of identification schemes, generic signatures and ring signatures. These results are well known, however, we generalise them in such a way that our technique can be used off-the-shelf for future applications. We note that less is more: as a side-effect of our less complex analysis, all our proofs are more precise; for example, we get a new proof of the forking lemma that is $2^{15}$ times more precise than the original result by Pointcheval and Stern. Finally, we give the first precise security analysis of Blum's coin flipping protocol with $k$-bit strings, as yet another example of the strength of our results.

In this paper, we show a general classification of cryptographically used elliptic and hyperelliptic curves which can be attacked by the Weil descent attack and index calculus algorithms. In particular, we classfy all the Weil restriction of these curves obtained by $(2,...,2)$ covering. Density analysis of these curves are shown. Explicit defintion equations of such weak curves are also provided.

Déchène has proposed generalized Jacobians as a source of groups for public-key cryptosystems based on the hardness of the Discrete Logarithm Problem (DLP). Her specific proposal gives rise to a group isomorphic to the semidirect product of an elliptic curve and a multiplicative group of a finite field. We explain why her proposal has no advantages over simply taking the direct product of groups. We then argue that generalized Jacobians offer poorer security and efficiency than standard Jacobians.

We introduce a new digital signature model, called conditionally verifiable signature (CVS), which allows a signer to specify and convince a recipient under what conditions his signature would become valid and verifiable; the resulting signature is not publicly verifiable immediately but can be converted back into an ordinary one (verifiable by anyone) after the recipient has obtained proofs, in the form of signatures/endorsements from a number of third party witnesses, that all the specified conditions have been fulfilled. A fairly wide set of conditions could be specified in CVS. The only job of the witnesses is to certify the fulfillment of a condition and none of them need to be actively involved in the actual signature conversion, thus protecting user privacy. It is guaranteed that the recipient cannot cheat as long as at least one of the specified witnesses does not collude. We formalize the concept of CVS and give a generic CVS construction based on any CPA-secure identity based encryption (IBE) scheme. Theoretically, we show that the existence of IBE with indistinguishability under a chosen plaintext attack (a weaker notion than the standard one) is necessary and sufficient for the construction of a secure CVS.\footnote{Due to page limit, some proofs are omitted here but could be found in the full version \cite{CB05ibecvs}.}

This paper presents a novel framework for the generic construction of hybrid encryption schemes which produces more efficient schemes than the ones known before. A previous framework introduced by Shoup combines a key encapsulation mechanism (KEM) and a data encryption mechanism (DEM). While it is sufficient to require both components to be secure against chosen ciphertext attacks (CCA-secure), Kurosawa and Desmedt showed a particular example of KEM that is not CCA-secure but can be securely combined with a specific type of CCA-secure DEM to obtain a more efficient, CCA-secure hybrid encryption scheme. There are also many other efficient hybrid encryption schemes in the literature that do not fit Shoup's framework. These facts serve as motivation to seek another framework. The framework we propose yields more efficient hybrid scheme, and in addition provides insightful explanation about existing schemes that do not fit into the previous framework. Moreover, it allows immediate conversion from a class of threshold public-key encryption to a hybrid one without considerable overhead, which may not be possible in the previous approach.

As more sensitive data is shared and stored by third-party sites on the Internet, there will be a need to encrypt data stored at these sites. One drawback of encrypting data, is that it can be selectively shared only at a coarse-grained level (i.e., giving another party your private key). We develop a new cryptosystem for fine-grained sharing of encrypted data that we call Key-Policy Attribute-Based Encryption (KP-ABE). In our cryptosystem, ciphertexts are labeled with sets of attributes and private keys are associated with access structures that control which ciphertexts a user is able to decrypt. We demonstrate the applicability of our construction to sharing of audit-log information and broadcast encryption. Our construction supports delegation of private keys which subsumes Hierarchical Identity-Based Encryption (HIBE).

In the CT-track of the 2006 RSA conference, a new multivariate public key cryptosystem, which is called the Medium Field Equation (MFE) multivariate public key cryptosystem, is proposed by Wang, Yang, Hu and Lai. We use the second order linearization equation attack method by Patarin to break MFE. Given a ciphertext, we can derive the plaintext within $2^{23}$ $\F_{2^{16}}$-operations, after performing once for any public key a computation of complexity less than $2^{52}$. We also propose a high order linearization equation (HOLE) attack on multivariate public key cryptosystems, which is a further generalization of the (first and second order) linearization equation (LE). This method can be used to attack extensions of the current MFE.

We investigate the impact of larger digit sets on the length of Double-Base Number system (DBNS) expansions. We present a new representation system called {\em extended DBNS} whose expansions can be extremely sparse. When compared with double-base chains, the average length of extended DBNS expansions of integers of size in the range 200--500 bits is approximately reduced by $20\%$ using one precomputed point, $30\%$ using two, and $38\%$ using four. We also discuss a new approach to approximate an integer $n$ by $d2^a3^b$ where $d$ belongs to a given digit set. This method, which requires some precomputations as well, leads to realistic DBNS implementations. Finally, a left-to-right scalar multiplication relying on extended DBNS is given. On an elliptic curve where operations are performed in Jacobian coordinates, improvements of up to $13\%$ overall can be expected with this approach when compared to window NAF methods using the same number of precomputed points. In this context, it is therefore the fastest method known to date to compute a scalar multiplication on a generic elliptic curve.

The Merkle signature scheme (MSS) is an interesting alternative for well established signature schemes such as RSA, DSA, and ECDSA. The security of MSS only relies on the existence of cryptographically secure hash functions. MSS has a good chance of being quantum computer resistant. In this paper, we propose CMSS, a variant of MSS, with reduced private key size, key pair generation time, and signature generation time. We demonstrate that CMSS is competitive in practice by presenting a highly efficient implementation within the Java Cryptographic Service Provider FlexiProvider. We present extensive experimental results and show that our implementation can for example be used to sign messages in Microsoft Outlook.

At CRYPTO 2004, Kurosawa and Desmedt presented a hybrid public-key encryption scheme that is chosen-ciphertext secure in the standard model. Until now it was unknown if the key-encapsulation part of the Kurosawa-Desmedt scheme by itself is still chosen-ciphertext secure or not. In this short note we answer this question to the negative, namely we present a simple chosen-ciphertext attack on the Kurosawa-Desmedt key encapsulation mechanism.

In cryptosystems based on elliptic curves over finite fields (ECC-systems), the most time-consuming operation is scalar multiplication. We focus on the NIST elliptic curves over prime fields. An implementation of scalar multiplication, developed by IBM Danmark A/S for test purposes, serves as a point of reference. In order to achieve maximal efficiency in an ECC-system, one must choose an optimal method for scalar multiplication and the best possible coordinate representation for the curve being used. We perform an analysis of known scalar multiplication methods. This analysis contains a higher degree of detail than existing publications on the subject and shows that the NAF$_w$ scalar multiplication method with precomputations in affine coordinates, intermediate doublings in Jacobian coordinates and additions in mixed coordinates is the optimal choice. We compare our scalar multiplication scheme with the one implemented by IBM and conclude that a substantial improvement of efficiency is achieved by using our scheme. We implement our efficient scheme and support our conclusions with timings of the implementations. Side channel attacks using power analysis is considered to be a major threat against the security of ECC-systems. Mathematical countermeasures exist but reduce the performance of the system. So far, no comparison of the countermeasures has been published. We perform such a comparison and conclude that if a sufficient amount of storage is available, a combination of side channel atomicity and scalar randomization should be used as a countermeasure. If storage is limited, countermeasures should be based on a combination of Montgomery's ladder algorithm and scalar randomization. We specify side channel atomic elliptic curve operations on the NIST elliptic curves in mixed coordinates. So far, no such specifications have been published. We develop an efficient and secure scalar multiplication scheme and conclude that this scheme is more efficient than the scheme used in the IBM implementation, which provides no security against side channel attacks. We implement our efficient, secure scheme and support our conclusions with timings of the implementations.

We describe algorithms for point multiplication on Koblitz curves using multiple-base expansions of the form $k = \sum \pm \tau^a (\tau-1)^b$ and $k= \sum \pm \tau^a (\tau-1)^b (\tau^2 - \tau - 1)^c.$ We prove that the number of terms in the second type is sublinear in the bit length of k, which leads to the first provably sublinear point multiplication algorithm on Koblitz curves. For the first type, we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of $\tau$-adic non-adjacent form expansions. We present details of an innovative FPGA implementation of our algorithm and performance data demonstrating the efficiency of our method.

The focus of this paper is to design an efficient and secure solution addressing the key escrow problem in ID-based signature schemes, i.e., the Private Key Generator (PKG) knows the user's private key, which damages the essential requirement--``non-repudiation" property of signature schemes. In this paper, we proposed two ID-based threshold signature schemes, which both reach Girault's trusted level 3, and in which there exists only one PKG in our ID-based threshold signature schemes. In particular, the second scheme has another good property: it does not require trusting any particular party at any time. Compared with the previous schemes, our schemes do not need to compute pairings, which make them be more efficient than those schemes. Furthermore, our ID-based signature schemes increase the availability of the signing agency and the difficulty for the adversary to learn the private key.

We introduce a simple primitive called Augmented Broadcast Encryption (ABE) that is sufficient for constructing broadcast encryption, traitor-tracing, and trace-and-revoke systems. These ABE-based constructions are resistant to an arbitrary number of colluders and are secure against adaptive adversaries. Furthermore, traitor tracing requires no secrets and can be done by anyone. These broadcast systems are designed for broadcasting to arbitrary sets of users. We then construct a secure ABE system for which the resulting concrete trace-and-revoke system has ciphertexts and private keys of size $\sqrt{N}$ where $N$ is the total number of users in the system. In particular, this is the first example of a fully collusion resistant broadcast system with sub-linear size ciphertexts and private keys that is secure against adaptive adversaries. The system is publicly traceable.

In most forward-secure signature constructions, a program that updates a user's private signing key must have full access to the private key. Unfortunately, these schemes are incompatible with several security architectures including Gnu Privacy Guard (GPG) and S/MIME, where the private key is encrypted under a user password as a ``second factor'' of security, in case the private key storage is corrupted, but the password is not. We introduce the concept of forward-secure signatures with untrusted update, where the key update can be performed on an encrypted version of the key. Forward secure signatures with untrusted update allow us to add forward security to signatures, while still keeping passwords as a second factor of security. We provide a construction that has performance characteristics comparable with the best existing forward-secure signatures. In addition, we describe how to modify the Bellare-Miner forward secure signature scheme to achieve untrusted update.

A main obstacle in manufacturing the TWIRL device for realizing the sieving step of the Number Field Sieve is the sophisticated chip layout. Especially the logic for logging and recovering large prime factors found during sieving adds significantly to the layout complexity. We describe a device building on the Elliptic Curve Method (ECM) that for parameters of interest enables the replacement of the complete logging part in TWIRL by an off-wafer postprocessing. The postprocessing is done in real time, leaving the total sieving time basically unchanged. The proposed device is an optimized ECM implementation building on curves chosen to cope with factor sizes as expected in the output of TWIRL. According to our preliminary analysis, for the relation collection step expected for a 1024-bit factorization our design is realizable with current fab technology at very moderate cost. The proposed ECM engine also finds the vast majority of the needed cofactor factorizations. In summary, we think the proposed device to enable a significant decrease of TWIRL's layout complexity and therewith its cost.

This paper presents a new software side-channel attack - enabled by the branch prediction capability common to all modern high-performance CPUs. The penalty payed (extra clock cycles) for a mispredicted branch can be used for cryptanalysis of cryptographic primitives that employ a data-dependent program flow. Analogous to the recently described cache-based side-channel attacks our attacks also allow an unprivileged process to attack other processes running in parallel on the same processor, despite sophisticated partitioning methods such as memory protection, sandboxing or even virtualization. We will discuss in detail several such attacks for the example of RSA, and experimentally show their applicability to real systems, such as OpenSSL and Linux. More specifically, we will present four different types of attacks, which are all derived from the basic idea underlying our novel side-channel attack. Moreover, we also demonstrate the strength of the branch prediction side-channel attack by rendering the obvious countermeasure in this context (Montgomery Multiplication with dummy-reduction) as useless. Although the deeper consequences of the latter result make the task of writing an efficient and secure modular expeonentiation (or scalar multiplication on an elliptic curve) a challenging task, we will eventually suggest some countermeasures to mitigate branch prediction side-channel attacks.

{\it This note was written in 1997 after a talk I gave at the sëminaire de complexitë et cryptographie at the École Normale Supërieure\footnote{http://www.di.ens.fr/~wwwgrecc/Seminaire/1996-97.html} After it was rejected at crypto97 I forgot it until a few colleagues of mine informed me that it could be of some interest to some researchers in the field of algorithmic and cryptography. Although I am not quite happy with the redaction of this note, I believe it is more fair not to improve nor correct it yet. So I leave it in its original state, including misprints. I just added this introductory paragraph. If need be, I will publish an updated version later.} We introduce the notion of hard homogeneous space (HHS) and briefly develop the corresponding theory. We show that cryptographic protocols based on the discrete logarithm problem have a counterpart for any hard homogeneous space. Indeed, the notion of hard homogeneous space is a more general and more natural context for these protocols. We exhibit conjectural hard homogeneous spaces independant from any discrete logarithm problem. They are based on complex multiplication theory. This shows the existence of schemes for authentication and key exchange that do not rely on the difficulty of computing dicrete logarithm in any finite group nor factoring integers. We show that the concept of HHS fits with class field theory to provide a unified theory for the already used discrete logarithm problems (on multiplicative groups of finite fields or rational points on elliptic curves) and the HHS we present here. We discuss a few algorithmic questions related to hard homogeneous spaces. The paper is looking for a wider point of view on the discrete logarithm problem both mathematically and cryptographically.

We describe the first efficient ring signature scheme secure, without random oracles, based on standard assumptions. Our ring signatures are based in bilinear groups. For $l$ members of a ring our signatures consist of $2l+2$ group elements and require $2l+3$ pairings to verify. We prove our scheme secure in the strongest security model proposed by Bender, Katz, and Morselli: namely, we show our scheme to be anonymous against full key exposure and unforgeable with respect to insider corruption. A shortcoming of our approach is that all the users' keys must be defined in the same group.

We show how to significantly speed-up the encryption portion of some public-key cryptosystems by the simple expedient of allowing a sender to maintain state that is re-used across different encryptions. In particular we present stateful versions of the DHIES and Kurosawa-Desmedt schemes that each use only one exponentiation to encrypt, as opposed to two and three respectively in the original schemes, yielding the fastest discrete-log based public-key encryption schemes known in the random-oracle and standard models respectively. The schemes are proven to meet an appropriate extension of the standard definition of IND-CCA security that takes into account novel types of attacks possible in the stateful setting.

KEM (Key Encapsulation Mechanism) was introduced by Shoup to formalize the asymmetric encryption specified for key distribution in ISO standards on public-key encryption. Shoup defined the ``semantic security (IND) against adaptively chosen ciphertext attacks (CCA2)'' as a desirable security notion of KEM. This paper introduces ''non-malleability (NM)'' of KEM, a stronger security notion than IND. We provide three definitions of NM, and show that these three definitions are equivalent. We then show that NM-CCA2 KEM is equivalent to IND-CCA2 KEM. That is, we show that NM is equivalent to IND under CCA2 attacks, although NM is stronger than IND in the definition (or under some attacks like CCA1). In addition, this paper defines the universally composable (UC) security of KEM and shows that NM-CCA2 KEM is equivalent to UC KEM.

We describe a new and practical identity-based key encapsulation mechanism that is secure in the standard model against chosen-ciphertext (CCA2) attacks. Since our construction is direct and not based on hierarchical identity-based encryption, it is more efficient than all previously proposed schemes. Furthermore, we give the first chosen-ciphertext secure identity-based key encapsulation mechanism with threshold key delegation and decryption in the standard model.

Frey proposed the idea of `disguising' an elliptic curve. This is a method to obtain a `black box' representation of a group. We adapt this notion to finite fields and tori and study the question of whether such systems are secure. Our main result is an algebraic attack which shows that it is not secure to disguise the torus $T_2$. We also show that some methods for disguising an elliptic curve are not secure. Finally, we present a method to disguise an elliptic curve which seems to resist our algebraic attack.

López-Dahab coordinates are usually the system of choice for implementations of elliptic curves over binary fields. We give new formulas for doubling which need one squaring less and one more addition. This leads to a speed-up for binary fields in polynomial basis representation.

Key issuing protocols deal with overcoming the two inherent problems: key escrow and secure channel requirement of the identity based cryptosystems. An efficient key issuing protocol enables the identity based cryptosystems to be more acceptable and applicable in the real world. We present a secure and efficient threshold key issuing protocol. In our protocol, neither KGC nor KPA can impersonate the users to obtain the private keys and thus it achieves the trust level III \cite{girault}. The protocol is secure against replay, man-in-the-middle and insider attacks.

The length-based approach is a heuristic for solving randomly generated equations in groups which possess a reasonably behaved length function. We describe several improvements of the previously suggested length-based algorithms, that make them applicable to Thompson's group with significant success rates. In particular, this shows that the Shpilrain-Ushakov public key cryptosystem based on Thompson's group is insecure, and suggests that no practical public key cryptosystem based on this group can be secure.

Pairings on elliptic curves have been used as cryptographic primitives for the development of new applications such as identity based schemes. For the practical applications, it is crucial to provide efficient and secure implementations of the pairings. There have been several works on efficient implementations of the pairings. However, the research for secure implementations of the pairings has not been thoroughly investigated. In this paper, we investigate vulnerability of the pairing used in some pairing based protocols against side channel attacks. We propose an efficient algorithm secure against such side channel attacks of the eta pairing using randomized projective coordinate systems for the pairing computation.

In this paper we propose the group of unitriangular matrices over a finite field as a non-abelian group and composition of inner, diagonal and central automorphisms as a group of automorphisms for the MOR cryptosystem.

We propose to verify the AKS algorithm identities not for sequential integers, but for integers which are sequentially squared. In that case a number of elements, for which the identities are valid, doubles.

At Eurocrypt 2004, Chen, Kudla and Paterson introduced the concept of {\it concurrent signatures}, which allows two parties to produce two ambiguous signatures until an extra piece of information (called {\it keystone}) is released by the initial signer. Once the keystone is released publicly, both signatures are binding to their true signers {\it concurrently}. At ICICS 2004, Susilo, Mu and Zhang further proposed {\it perfect concurrent signatures} to strengthen the ambiguity of concurrent signatures. That is, even the both signers are known having issued one of the two ambiguous signatures, any third party is still unable to deduce who signed which signature, different from Chen et al.'s scheme. However, this paper points out that Susilo et al.'s two perfect concurrent signatures are actually {\it not} concurrent signatures. Specifically, we identify an attack that enables the initial signer to release a carefully prepared keystone that binds the matching signer's signature, but not the initial signer's. Therefore, both of their two schemes are {\it unfair} for the matching signer. Moreover, we present a simple but effective way to avoid this attack such that the improved schemes are truly perfect concurrent signatures.

Pairing-based cryptosystems have been developing very fast in the last few years. The efficiencies of the cryptosystems are determined by the computation of the Tate pairing. In this paper a new efficient algorithm based on double-base chain for computing the Tate pairing is proposed for odd characteristic $p>3$. The inherent sparseness of double-base number system reduces the computational cost for computing the Tate pairing evidently. It is $9\%$ faster than the previous fastest method for MOV degree k=6.

We generalize the selective-ID security model for HIBE by introducing two new security models. Broadly speaking, both these models allow the adversary to commit to a set of identities and in the challenge phase choose any one of the previously committed identities. Two constructions of HIBE are presented which are secure in the two models. Further, we show that the HIBEs can be modified to obtain a multiple receiver IBE which is secure in the selective-ID model without the random oracle assumption.

Signature schemes with message recovery have been wildly investigated a decade ago in the literature, but the first ID-based signature with message recovery goes out into the world until 2005. In this paper, we first point out and revise one little but important problem which occurs in the previous ID-based signature with message recovery scheme. Then, by completely different setting, we propose a new ID-based signature scheme with message recovery. Our scheme is much more efficient than the previous scheme. In our scheme (as well as other signature schemes with message recovery), the message itself is not required to be transmitted together with the signature, it turns out to have the least data size of communication cost comparing with generic (not short) signature schemes. Although the communication overhead is still larger than Boneh et al. 's short signature (which is not ID-based), the computational cost of our scheme is more efficient than Boneh et al. 's scheme in the verification phase. We will also prove that the proposed scheme is provably secure in the random oracle model under CDH Assumption.

We construct the first several efficient threshold ring signatures (TRS) without random oracles. Specializing to a threshold of one, they are the first several efficient ring signatures without random oracles after the only earlier instantiation of Chow, Liu, Wei, and Yuen. Further specializing to a ring of just one user, they are the short (ordinary) signatures without random oracles summarized in Wei and Yuen. We also construct the first hierarchical identity-based threshold ring signature without random oracles. The signature size is $O(n\lambda_s)$ bits, where $\lambda_s$ is the security parameter and $n$ is the number of users in the ring. Specializing to a threshold of one, it is the first hierarchical identity-based ring signature without random oracles. Further specializing to a ring of one user, it is the constant-size hierarchical identity-based signature (HIBS) without random oracles in Yuen-Wei - the signature size is $O(\lambda_s)$ bits which is independent of the number of levels in the hierarchy.

RSA type public key cryptosystems based on the Pell's equation are proposed in the honor of an Indian mathematician Brahmgupta who studied Pell's equation long before European mathematicians came to know about it. Three RSA type schemes are proposed, first two are not semantically secure where as the other two schemes are semantically secure. The decryption speed of the proposed schemes is about two times as fast as RSA for a 2 log n-bit message. It is shown that the proposed schemes are more secure than the RSA scheme when purely common plaintexts are encrypted in the broadcast application and are as secure as the RSA scheme against ciphertext attack. In addition the proposed schemes are also secure against partially known plaintext attack. First two are not semantically secure but the third one is semantically secure.

We present an identity-based cryptosystem that features fully anonymous ciphertexts and hierarchical key delegation. We give a proof of security in the standard model, based on the mild Decision Linear complexity assumption in bilinear groups. The system is efficient and practical, with small ciphertexts of size linear in the depth of the hierarchy. Applications include search on encrypted data, fully private communication, etc. Our results resolve two open problems pertaining to anonymous identity-based encryption, our scheme being the first to offer provable anonymity in the standard model, in addition to being the first to realize fully anonymous HIBE at all levels in the hierarchy.

We describe a new encryption technique that is secure in the standard model against adaptive chosen ciphertext (CCA2) attacks. We base our method on two very efficient Identity-Based Encryption (IBE) schemes without random oracles due to Boneh and Boyen, and Waters. Unlike previous CCA2-secure cryptosystems that use IBE as a black box, our approach is endogenous, very simple, and compact. It makes direct use of the underlying IBE structure, and requires no cryptographic primitive other than the IBE scheme itself. This conveys several advantages. We achieve shorter ciphertext size than the best known instantiations of the other methods, and our technique is as efficient as the Boneh and Katz method (and more so than that of Canetti, Halevi, and Katz). Further, our method operates nicely on hierarchical IBE, and since it allows the validity of ciphertexts to be checked publicly, it can be used to construct systems with non-interactive threshold decryption. In this paper we describe two main constructions: a full encryption system based on the Waters adaptive-ID secure IBE, and a KEM based on the Boneh-Boyen selective-ID secure IBE. Both systems are shown CCA2-secure in the standard model, the latter with a tight reduction. We discuss several uses and extensions of our approach, and draw comparisons with other schemes that are provably secure in the standard model.

A new general mathematical problem, suitable for public-key cryptosystems, is proposed: morphism computation in a category of Abelian groups. In connection with elliptic curves over finite fields, the problem becomes the following: compute an isogeny (an algebraic homomorphism) between the elliptic curves given. The problem seems to be hard for solving with a quantum computer. ElGamal public-key encryption and Diffie-Hellman key agreement are proposed for an isogeny cryptosystem. The paper describes theoretical background and a public-key encryption technique, followed by security analysis and consideration of cryptosystem parameters selection. A demonstrative example of encryption is included as well.

In this paper, we revisit the problem of computing the kernel of a separable isogeny of degree $\ell$ between two elliptic curves defined over a finite field $\GF{q}$ of characteristic $p$. We describe an algorithm the asymptotic time complexity of which is equal to $\SoftO(\ell^2(1+\ell/p)\log q)$ bit operations. This algorithm is particularly useful when $\ell > p$ and as a consequence, we obtain an improvement of the complexity of the SEA point counting algorithm for small values of $p$. More precisely, we obtain a heuristic time complexity $\SoftO(\log^{4} q)$ and a space complexity $O(\log^{2} q)$, in the previously unfavorable case where $p \simeq \log q$. Compared to the best previous algorithms, the memory requirements of our SEA variation are smaller by a $\log^2 q$ factor.

In this paper, we present some more results about the security of the Kurosawa-Desmedt encryption scheme and a variant of it. We prove that after a modification, those schemes are secure against adaptive chosen-ciphertext attack not only under the decisional Diffie-Hellman assumption in standard model as before but also under the computational Diffie-Hellman assumption in the random oracle model. These results ensure that both the Kurosawa-Desmedt scheme and the variant have similar security merits as the Cramer-Shoup encryption scheme, which is proposed as a standard.

Many research papers in pairing based cryptography treat pairings as a ``black box''. These papers build cryptographic schemes making use of various properties of pairings. If this approach is taken, then it is easy for authors to make invalid assumptions concerning the properties of pairings. The cryptographic schemes developed may not be realizable in practice, or may not be as efficient as the authors assume. The aim of this paper is to outline, in as simple a fashion as possible, the basic choices that are available when using pairings in cryptography. For each choice, the main properties and efficiency issues are summarized. The paper is intended to be of use to non-specialists who are interested in using pairings to design cryptographic schemes.

In this paper we give an overview of some of the cryptographic applications which were derived from the proposal of R.J. McEliece to use error correcting codes for cryptographic purposes. Code based cryptography is an interesting alternative to number theoretic cryptography. Many basic cryptographic functions like encryption, signing, hashing, etc. can be realized using code theoretic concepts. In this paper we briefly show how to correct errors in transmitted data by employing Goppa codes and describe possible applications to public key cryptography. The main focus of this paper is to provide detailed insight into the state of art of cryptanalysis of the McEliece cryptosystem and the effect on different cryptographic applications. We conclude, that for code based cryptography a public key of $88$KB offers sufficient security for encryption, while we need a public key of at least $597$KB for secure signing.

This paper proposes a new ID-based proxy signature scheme based on the bilinear pairings. The number of paring operation involved in the verification procedure of our scheme is only one, so our scheme is more efficient comparatively. The new scheme can be proved secure with the hardness assumption of the k-Bilinear Diffie-Hellman Inverse problem, in the random oracle model.

Signcryption schemes aim to provide all of the advantages of simultaneously signing and encrypting a message. Recently, Dent and Bjørstad investigated the possibility of constructing provably secure signcryption schemes using hybrid KEM-DEM techniques. We build on this work by showing that more efficient insider secure hybrid signcryption schemes can be built using Tag-KEMs. To prove the effectiveness of this construction, we will provide several examples of secure signcryption Tag-KEMs, including a brand new construction based on the Chevallier-Mames signature scheme which has the tightest known security reductions for both confidentiality and unforgeability.

Two variants of CA-based public key authentication framework are proposed in this paper. The one is termed as public key cryptosystem without certificate management center (PKCwCMC) and the other is termed as proxy signature based authentication framework (PS-based AF). Moreover, we give an implementation of the former based on quadratic residue theory and an implementation of the latter from RSA. Both of the two variants can be looked as lite-CA based authentication frameworks since the workload and deployment of CAs in these systems are much lighter and easier than those of in the traditional CA-based PKC.

In this paper we examine the security criteria for a KEM and a DEM that are su±cient for the overall hybrid encryption scheme to be plaintext-aware in the standard model. We apply this theory to the Cramer-Shoup hybrid scheme acting on ¯xed length messages and deduce that the Cramer-Shoup scheme is plaintext-aware in the standard model. This answers a previously open conjecture of Bellare and Palacio on the existence of plaintext-aware encryption schemes.

The only known construction of identity-based signatures that can be proven secure in the standard model is based on the approach of attaching certificates to non-identity-based signatures. This folklore construction method leads to schemes that are somewhat inefficient and leaves open the problem of finding more efficient direct constructions. We present the first such construction. Our scheme is obtained from a modification of Waters' recently proposed identity-based encryption scheme. It is computationally efficient and the signatures are short. The scheme's security is proven in the standard model and rests on the hardness of the computational Diffie-Hellman problem in groups equipped with a pairing.

Identity based cryptosystem simplifies the key management and revocation problem. Here we propose an Identity Based Strong Designated Verifier Signature (IBSDVS) scheme using bilinear pairings. The Designated Verifier Signature scheme described in [10] is identity based but it suffers from the deligatability as pointed out in [4]. We analyse the security of the scheme and show that the problem of delegatability does not exist in our scheme.

In this paper we introduce a new approach of constructing time capsule signature. Our new construction captures the basic requirements defined by dodis \emph{et al.}, and it is also very straightforward and flexible. The time capsule signature provides an elegant way to produce a ``future signature" that becomes valid from a specific future time $t$, when a trusted third party (called \textit{Time Server}) publishes some trapdoor information associated with the time $t$. It also has many other advantages. Our work includes a developed security model of time capsule signature, a novel way of construction based on the bipartite ring signature, which is proven secure in the random oracle model and a concrete realization of the scheme.

In this paper we study security definitions for authenticated key exchange (AKE) protocols. We observe that there are several families of attacks on AKE protocols that lie outside the boundary of the current class of security definitions. In an attempt to bring these attacks within the scope of analysis we extend the AKE security definition to provide greater powers to the adversary. We provide a general framework for defining AKE security, which we call strong AKE security, such that existing security definitions occur as instances of the framework. We then introduce NAXOS, a new two-pass AKE protocol, and prove that it is secure in this stronger definition. In addition, we formulate a notion of ephemeral secret key which captures all ephemeral information used in session establishment. We demonstrate the importance of this formulation by showing that a secure AKE protocol SIG-DH can become vulnerable when instantiated with signature schemes which are insecure against revelation of the secret random bits used in the signature generation.

Signcryption is a new cryptographic primitive that simultaneously fulfills both the functions of signature and encryption. The definition of generalized signcryption is proposed in the paper firstly. Generalized signcryption has a special feature that provides confidentiality or authenticity separately under the condition of specific inputs. So it is more useful than common ones. Based on ECDSA, a signcryption scheme called ECGSC is designed. It will be equivalent to an AtE(OTP$,MAC) encryption scheme or ECDSA when one of party is absent. A third party can verify the signcryption text publicly in the method of ECDSA. Security properties are proven based on Random Oracle mode: confidentiality (CUF-CPA), unforgeability (UF-CMA) and non-repudiation. Compared with the others, ECGSC presents a 78% reduction in computational cost for typical security parameters for high level security applications.

The fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication. Based on the double-base chain representation of scalar using powers of 2 and 3, we propose a new representation with powers of ½ and 3 instead. Thus the efficient point halving operation can be incorporated in the new double-base chain to achieve fast scalar multiplication. Experimental results show that our approach leads to a lower complexity which contributes to the efficient implementation of elliptic curve cryptosystems.

A cover-free family is a well-studied combinatorial structure that has many applications in computer science and cryptography. In this paper, we propose a new public key traitor tracing scheme based on cover-free families. The new traitor tracing scheme is similar to the Boneh-Franklin scheme except that in the Boneh-Franklin scheme, decryption keys are derived from Reed-Solomon codes while in our case they are derived from a cover-free family. This results in much simpler and faster tracing algorithms for single-key pirate decoders, compared to the tracing algorithms of Boneh-Franklin scheme that use Berlekamp-Welch algorithm. Our tracing algorithms never accuse innocent users and identify all traitors with overwhelming probability.

We define key privacy for IBE systems in terms of two properties, indistinguishability under chosen identity attack, and indistinguishability under chosen key generator attack. Further, we show that the BasicIdent system in the Boneh/Franklin IBE has these properties under chosen plaintext attack.

Public key encryption with keyword search (PEKS) enables user Alice to send a secret key $T_W$ to a server that will enable the server to locate all encrypted messages containing the keyword $W$, but learn nothing else. In this paper, we propose a new PKES scheme based on pairings. There is no pairing operation involved in the encryption procedure. Then, we provide further discussion on removing secure channel from PKES, and present an efficient secure channel free PKES scheme. Our two new schemes can be proved secure in the random oracle model, under the appropriate computational assumptions.

We define the weight of an integer $N$ to be the smallest $w$ such that $N$ can be represented as $\sum_{i=1}^w \epsilon_i 2^{c_i}$, with $\epsilon_1,\ldots,\epsilon_w\in\{1,-1\}$. Since arithmetic modulo a prime of low weight is particularly efficient, it is tempting to use such primes in cryptographic protocols. In this paper we consider the difficulty of the discrete logarithm problem modulo a prime $N$ of low weight, as well as the difficulty of factoring an integer $N$ of low weight. We describe a version of the number field sieve which handles both problems. Our analysis leads to the conjecture that, for $N\to\infty$ with $w$ fixed, the worst-case running time of the method is bounded above by ${\rm exp}((c+o(1))(\log\,N)^{1/3}(\log\log\,N)^{2/3})$ with $c<((32/9)(2w-3)/(w-1))^{1/3}$ and below by the same expression with $c=(32/9)^{1/3}((\sqrt{2}w-2\sqrt{2}+1)/(w-1))^{2/3}.$ It also reveals that on average the method performs significantly better than it does in the worst case. We consider all the examples given in a recent paper of Koblitz and Menezes and demonstrate that in every case but one, our algorithm runs faster than the standard versions of the number field sieve.

This paper proposes a new efficient signature scheme from bilinear maps that is secure in the standard model (i.e., without the random oracle model). Our signature scheme is more effective in many applications (e.g., blind signatures, group signatures, anonymous credentials etc.) than the existing secure signature schemes in the standard model. As typical applications of our signature scheme, this paper presents efficient blind signatures and partially blind signatures that are secure in the standard model. Here, partially blind signatures are a generalization of blind signatures (i.e., blind signatures are a special case of partially blind signatures) and have many applications including electronic cash and voting. Our blind signature scheme is much more efficient than the existing secure blind signature schemes in the standard model such as the Camenisch-Koprowski-Warinsch and Juels-Luby-Ostrovsky schemes, and is also almost as efficient as the most efficient blind signature schemes whose security has been analyzed heuristically or in the random oracle model. Our partially blind signature scheme is the first one that is secure in the standard model and it is very efficient (as efficient as our blind signatures). The security proof of our blind and partially blind signature schemes requires the 2SDH assumption, a variant of the SDH assumption introduced by Boneh and Boyen, and the 2SDH-IND assumption. This paper also presents an efficient way to convert our (partially) blind signature scheme in the standard model to a scheme secure for a concurrent run of users in the common reference string (CRS) model. Finally, we present a blind signature scheme based on the Waters signature scheme.

We present the first aggregate signature, the first multisignature, and the first verifiably encrypted signature provably secure without random oracles. Our constructions derive from a novel application of a recent signature scheme due to Waters. Signatures in our aggregate signature scheme are sequentially constructed, but knowledge of the order in which messages were signed is not necessary for verification. The aggregate signatures obtained are shorter than Lysyanskaya et~al. sequential aggregates and can be verified more efficiently than Boneh et~al. aggregates. We also consider applications to secure routing and proxy signatures.

Let $N=pq$ be an RSA modulus where $p$, $q$ are large primes of the same bitsize. We study the class of the public exponents $e$ for which there exist an integer $m$ with $1\leq m\leq {\log{N}\over \log{32}}$ and small integers $u$, $X$, $Y$ and $Z$ satisfying $$(e+u)Y^m-\psi(N)X^m=Z,$$ where $\psi(N)=(p+1)(q-1)$. First we show that these exponents are of improper use in RSA cryptosystems. Next we show that their number is at least $O\left(mN^{{1\over 2}+{\a\over m}-\a-\e}\right)$ where $\a$ is defined by $N^{1-\a}=\psi(N)$.

We present the first efficient group signature scheme that is provably secure without random oracles. We achieve this result by combining provably secure hierarchical signatures in bilinear groups with a novel adaptation of the recent Non-Interactive Zero Knowledge proofs of Groth, Ostrovsky, and Sahai. The size of signatures in our scheme is logarithmic in the number of signers; we prove it secure under the Computational Diffie-Hellman and the Subgroup Decision assumptions in the model of Bellare, Micciancio, and Warinshi, as relaxed by Boneh, Boyen, and Shacham.

Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree $k \leqslant 6$. More general methods produce curves over $\F_p$ where the bit length of $p$ is often twice as large as that of the order $r$ of the subgroup with embedding degree $k$; the best published results achieve $\rho \equiv \log(p)/\log(r) \sim 5/4$. In this paper we make the first step towards surpassing these limitations by describing a method to construct elliptic curves of prime order and embedding degree $k = 12$. The new curves lead to very efficient implementation: non-pairing cryptosystem operations only need $\F_p$ and $\F_{p^2}$ arithmetic, and pairing values can be compressed to one \emph{sixth} of their length in a way compatible with point reduction techniques. We also discuss the role of large CM discriminants $D$ to minimize $\rho$; in particular, for embedding degree $k = 2q$ where $q$ is prime we show that the ability to handle $\log(D)/\log(r) \sim (q-3)/(q-1)$ enables building curves with $\rho \sim q/(q-1)$.

A parity checking-styled Weil sum algorithm is presented for a general class of the univariate polynomials which fully characterize a system of $n$ polynomials in $n$ variables over $F_{2}$. The previously known proof methods of explicit Weil sum evaluation of Dembowski-Ostrom polynomials are extended to general case. The algorithm computes the absolute values of the Weil sums of the generic central polynomials in MQ problem.

In number theoretic cryptography there is always the problem of scaling-up security to a higher level. This usually means increasing the size of the modulus, from, say 1024 bits to 2048 bits. In pairing-based cryptography however another option is available, keeping the modulus constant and increasing instead the embedding degree. This has a big potential advantage in smart-card and embedded applications -- security can be scaled up while continuing to use the same sized calculations. For example a cryptographic co-processor which does 512-bit modular multiplications can be directly re-used in the higher security setting. Here we investigate the scaling-up issue in the context of prime characteristic non-supersingular elliptic curves. We also confirm the observation that at higher levels of security a slightly modified Weil pairing becomes more efficient than the Tate pairing.

At the Asiacrypt 2003 conference Billet and Gilbert introduce a block cipher, which, to quote them, has the following paradoxical traceability properties: it is computationally easy to derive many equivalent distinct descriptions of the same instance of the block cipher; but it is computationally difficult, given one or even up to $k$ of them, to recover the so-called meta-key from which they were derived, or to find any additional equivalent description, or more generally to forge any new untraceable description of the same instance of the block cipher. Their construction relies on the Isomorphism of Polynomials (IP) problem. We here show how to strengthen this construction against algebraic attacks by concealing the underlying IP problems. Our modification is such that our description of the block cipher now does not give the expected results all the time and parallel executions are used to obtain the correct value.

The security and performance of pairing based cryptography has provoked a large volume of research, in part because of the exciting new cryptographic schemes that it underpins. We re-examine how one should implement pairings over ordinary elliptic curves for various practical levels of security. We conclude, contrary to prior work, that the Tate pairing is more efficient than the Weil pairing for all such security levels. This is achieved by using efficient exponentiation techniques in the cyclotomic subgroup backed by efficient squaring routines within the same subgroup.

The paper analyzes a new public key cryptosystem whose security is based on a matrix version of the discrete logarithm problem over an elliptic curve. It is shown that the complexity of solving the underlying problem for the proposed system is dominated by the complexity of solving a fixed number of discrete logarithm problems in the group of an elliptic curve. Using an adapted Pollard rho algorithm it is shown that this problem is essentially as hard as solving one discrete logarithm problem in the group of an elliptic curve.

We proposed the concept, piece in hand (soldiers in hand) matrix and have developed the framework based on the concept so far. The piece in hand matrix is a general concept which can be applicable to any type of multivariate public key cryptosystems to enhance their security. In this paper, we make improvements in the PH matrix method as follows. (i) In the PH matrix method, an arbitrary number of additional variables can be introduced to the random polynomial term in the public key, which is eliminated by the multiplication of the PH matrix to the public key in the decryption. Thus these additional variables enables the public key to have more than one solution, and therefore increases the difficulty to solve the public key. We show, in an experimental manner, that the PH matrix method improved in this way is secure even against the Gröbner basis attack. (ii) In the nonlinear PH matrix method proposed previously, the degree of polynomials in the public key is more than two, and this may cause an undesirable increase in the length of the public key. In this paper, we propose a nonlinear PH matrix method, where the degree of the public key is kept the same as the degree of the public key of the original cryptosystem, which is normally two.

We construct three new signatures and prove their securities without random oracles. They are motivated, respectively, by Boneh and Boyen's, Zhang, et al.'s, and Camenisch and Lysyanskaya's signatures without random oracles. The first two of our signatures are as short as Boneh and Boyen's (resp. Zhang, et al.'s} state-of-the-art short signatures. Our third signature is reducible to a modified LRSW Assumption but without their hypothesized external signing oracle. New and interesting variants of the q-SDH Assumption, the q-SR (Square Root) Assumption are also presented. New and independently interesting proof techniques extending the two-mode technique of Boneh and Boyen are used, including a combined three-mode simulation and rewinding in the standard model.

Recently, Al-Riyami and Paterson proposed a generic conversion from CL-PKE (Certificateless Public Key Encryption) to CBE (Certificate Based Encryption) and claimed that the derived CBE scheme is secure and even more efficient than the original scheme of Gentry. In this paper, we show that their conversion is wrong due to the flaw of the security proof. It leads the new concrete CBE scheme by Al-Riyami and Paterson to be invalidated. In addition, our result supports the impossibility to relate both notions in any directions.

This paper aims at introducing generalized Jacobians as a new candidate for discrete logarithm (DL) based cryptography. The motivation for this work came from the observation that several practical DL-based cryptosystems, such as ElGamal, the Elliptic and Hyperelliptic Curve Cryptosystems, XTR, LUC as well as CEILIDH can all naturally be reinterpreted in terms of generalized Jacobians. However, usual Jacobians and algebraic tori are thus far the only generalized Jacobians implicitly utilized in cryptography. In order to go one step further, we here study the simplest nontrivial generalized Jacobians of an elliptic curve. In this first of a series of articles, we obtain explicit formulae allowing to efficiently perform arithmetic operations in these groups. This work is part of our doctoral dissertation, where security aspects are considered in depth. As a result, these groups thus provide the first concrete example of semi-abelian varieties suitable for DL-based cryptography.

Kurosawa-Desmedt encryption scheme is a variation of Cramer-Shoup encryption schemes, which are the first practical schemes secure against adaptive chosen ciphertext attack in standard model. We introduce a variant of Kurosawa-Desmedt encryption scheme, which is not only secure against adaptive chosen ciphertext attack but also slightly more efficient than the original version.

Authenticated encryption schemes used in order to send one message to one recipient have received considerable attention in the last years. We investigate the case of schemes, we call authenticated $\mathtt{1{\to}n}$ schemes, that allow one to encrypt efficiently in a public-key setting a message for several, say $n$, recipients in an authenticated manner. We propose formal security definitions for such schemes that work also for $n=1$ and which are stronger and/or more general than those currently proposed. We then present a flexible mode of operation that transforms any $\mathtt{1{\to}1}$ authenticated encryption scheme working on small messages into a $\mathtt{1{\to}n}$ authenticated encryption scheme working on longer messages. We show that it allows the construction of efficient $\mathtt{1{\to}n}$ schemes that are proved secure for the strongest security notion.

We present a general framework for constructing families of elliptic curves of prime order with prescribed embedding degree. We demonstrate this method by constructing curves with embedding degree k = 10, which solves an open problem posed by Boneh, Lynn, and Shacham. We show that our framework incorporates existing constructions for k = 3, 4, 6, and 12, and we give evidence that the method is unlikely to produce infinite families of curves with embedding degree k > 12.

We propose a new class of key establishment schemes which are based on geometric generalizations of the classical Diffie-Hellman. The simplest of our schemes – based on the geometry of the unit circle – uses only multiplication of rational numbers by integers and addition of rational numbers in its key creation. Its first computer implementation works significantly faster than all known implementations of Diffie-Hellman. Preliminary estimations show that our schemes are resistant to attacks. This resistance follows the pattern of the discrete logarithm problem and hardness of multidimensional lattice problems

We present a Hierarchical Identity Based Encryption (HIBE) system where the ciphertext consists of just three group elements and decryption requires only two bilinear map computations, independent of the hierarchy depth. Encryption is as efficient as in other HIBE systems. We prove that the scheme is selective-ID secure in the standard model and fully secure in the random oracle model. Our system has a number of applications: it gives very efficient forward secure public key and identity based cryptosystems (where ciph ertexts are short), it converts the NNL broadcast encryption system into an efficient public key broadcast system, and it provides an efficient mechanism for encrypting to the future. The system also supports limited delegation where users can be given restricted private keys that only allow delegation to certain descendants. Sublinear size private keys can also be achieved at the expense of some ciphertext expansion.

A nominative signature scheme allows a nominator (or signer) and a nominee (or verifier) to jointly generate and publish a signature in such a way that \emph{only} the nominee can verify the signature and if necessary, \emph{only} the nominee can prove to a third party that the signature is valid. In a recent work, Huang and Wang proposed a new nominative signature scheme which, in addition to the above properties, \emph{only} allows the nominee to convert a nominative signature to a publicly verifiable one. In ACISP 2005, Susilo and Mu presented several algorithms and claimed that these algorithms can be used by the nominator to verify the validity of a published nominative signature, show to a third party that the signature is valid, and also convert the signature to a publicly verifiable one, all \emph{without} any help from the nominee. In this paper, we point out that Susilo and Mu's attacks are actually \emph{incomplete} and {\it inaccurate}. In particular, we show that there exists no efficient algorithm for a nominator to check the validity of a signature if this signature is generated by the nominator and the nominee {\it honestly} and the Decisional Diffie-Hellman Problem is hard. On the other hand, we point out that the Huang-Wang scheme is indeed {\it insecure}, since there is an attack that allows the nominator to generate valid nominative signatures alone and prove the validity of such signatures to a third party.

We show that in applications that use the Diffie-Hellman (DH) transform but take care of hashing the DH output (as required, for example, for secure DH-based encryption and key exchange) the usual requirement to work over a DDH group (i.e., a group in which the Decisional Diffie-Hellman assumption holds) can be relaxed to only requiring that the DH group contains a large enough DDH subgroup. In particular, this implies the security of (hashed) Diffie-Hellman over non-prime order groups such as $Z_p^*$. Moreover, our results show that one can work directly over $Z_p^*$ without requiring any knowledge of the prime factorization of $p-1$ and without even having to find a generator of $Z_p^*$. These results are obtained via a general characterization of DDH groups in terms of their DDH subgroups, and a relaxation (called $t$-DDH) of the DDH assumption via computational entropy. We also show that, under the short-exponent discrete-log assumption, the security of the hashed Diffie-Hellman transform is preserved when replacing full exponents with short exponents.

We introduce the concept of identity based key encapsulation to multiple parties (mID-KEM), and define a security model for it. This concept is the identity based analogue of public key KEM to multiple parties. We also analyse possible mID-KEM constructions, and propose an efficient scheme based on bilinear pairings. We prove our scheme secure in the random oracle model under the Gap Bilinear Diffie-Hellman assumption.

In this thesis we consider homomorphic cryptosystems and their applications. Homomorphic cryptosystems allow for computations on encrypted data. We prove that the search for an algebraically homomorphic scheme can be reduced to the search of a homomorphic scheme on a special non-abelian group. Furthermore, we focus on a special application: computing with encrypted functions and data, respectively. For this application we develop an improved protocol that is efficient for functions that are computable by polynomial branching programs. Finally, we generalise the elliptic curve Paillier scheme by S. Galbraith in order to construct a threshold version of it. For this threshold scheme we develop several Sigma-protocols. Using these protocols we apply our threshold scheme on multiparty computations, electronic voting and commitment schemes.

Multivariate Quadratic public key schemes have been suggested back in 1985 by Matsumoto and Imai as an alternative for the RSA scheme. Since then, several other schemes have been proposed, for example Hidden Field Equations, Unbalanced Oil and Vinegar schemes, and Stepwise Triangular Schemes. All these schemes have a rather large key space for a secure choice of parameters. Surprisingly, the question of equivalent keys has not been discussed in the open literature until recently. In this article, we show that for all basic classes mentioned above, it is possible to reduce the private --- and hence the public --- key space by several orders of magnitude. For the Matsumoto-Imai scheme, we are even able to show that the reductions we found are the only ones possible, i.e., that these reductions are tight. While the theorems developed in this article are of independent interest themselves as they broaden our understanding of Multivariate Quadratic public key systems, we see applications of our results both in cryptanalysis and in memory efficient implementations of MQ-schemes.

We present a modification of the well-known hidden number problem (HNP) which we refer to as a one-time HNP (OT-HNP). We also present an algorithm for solving such a problem together with its formal analysis. We show then that carefully designed instances of OT-HNP can be used to break certain flawed implementations of public key schemes efficiently. We work, for instance, with Nguyen’s attack on El Gamal’s signature scheme in the GNU Privacy Guard of version 1.2.3. The technique employed there was not based on HNP, since it was supposed that more than one signature would be necessary, which seemed to be a wastage. We will see, however, that by using OT-HNP one signature is still far enough, while retaining certain elegance of the HNP approach. We also present an experimental confirmation of this result.

In this article we show that Shim¡¦s new ID-based tripartite multiple-key agreement protocol still suffers from the impersonation attack, a malicious user can launch an impersonation attack on their protocol.

The paper proposes a new scheme of authenticated encryption that is either publicly verifiable or not publicly verifiable depending on the quantity of information the recipient released. This property would give recipient much flexibility in many applications. This scheme combines the ElGamal encryption with Schnorr signature. Considering the security goal of signature, the resultant scheme is at least as secure as that of the combined signature scheme. The security goal of encryption is examined under the chosen ciphertext attack, it is proven directly related to the security of signature. Furthermore, this new scheme is also secure against one-more-decryption attack. This novel security goal may be valuable in the applications of private information retrieval.

In this paper, we show explicitly the classes of elliptic and hyperelliptic curves of low genera defined over extension fields, which have weak coverings, i.e. their Weil restrictions can be attacked by either index calculus attacks to hyperelliptic curves or Diem's recent attack to non-hyperelliptic curves. In particular, we show how to construct such coverings from these curves and analyze density of the curves for them such construction is possible.

Multivariate quadratic systems can be used to construct both secure and efficient public key schemes. In this article, we introduce the necessary mathematical tools to deal with multivariate quadratic systems, present an overview of important schemes known so far and outline how they fit into a taxonomy of only four basic schemes and some generic modifiers. Moreover, we suggest new constructions not previously considered. In this context, we propose some open problems and new research directions in the field of multivariate quadratic schemes.

Ring signatures, first introduced by Rivest, Shamir, and Tauman, enable a user to sign a message so that a ring of possible signers (of which the user is a member) is identified, without revealing exactly which member of that ring actually generated the signature. In contrast to group signatures, ring signatures are completely ``ad-hoc'' and do not require any central authority or coordination among the various users (indeed, users do not even need to be aware of each other); furthermore, ring signature schemes grant users fine-grained control over the level of anonymity associated with any particular signature. This paper has two main areas of focus. First, we examine previous definitions of security for ring signature schemes and suggest that most of these prior definitions are too weak, in the sense that they do not take into account certain realistic attacks. We propose new definitions of anonymity and unforgeability which address these threats, and give separation results proving that our new notions are strictly stronger than previous ones. Second, we show the first constructions of ring signature schemes in the standard model. One scheme is based on generic assumptions and satisfies our strongest definitions of security. Two additional schemes are more efficient, but achieve weaker security guarantees and more limited functionality.

In this paper, we discuss non-interactive updating of decryption keys in identity-based encryption (IBE). IBE is a public key cryptosystem where a public key is an arbitrary string. In practice, key revocation is a necessary and inevitable process and IBE is no exception when it comes to having to manage revocation of decryption keys without losing its merits in efficiency. Our main contribution of this paper is to propose novel constructions of IBE where the decryption key can be renewed without having to make changes to its public key, i.e. user's identity. We achieve this by tactfully extending the hierarchical IBE (HIBE). Regarding security, we address semantic security against adaptive chosen cipher-text attack for a very strong attack environment that models all possible types of key exposures in the random oracle model. Straightforward extension of the HIBE, however, does not achieve our goal and such scheme is completely insecure under our attack model. In addition to this, we show method of constructing (partially collusion resistant) HIBE from arbitrary IBE in the random oracle model. By combining these results, we can construct an IBE with non-interactive key update from only an arbitrary IBE.

In this paper we present a new key establishment protocol based on the decomposition problem in non-commutative groups which is: given two elements w, w_1 of the platform group G and two subgroups A, B of G (not necessarily distinct), find elements a in A, b in B such that w_1 = a w b. Here we introduce two new ideas that improve the security of key establishment protocols based on the decomposition problem. In particular, we conceal (i.e., do not publish explicitly) one of the subgroups A, B, thus introducing an additional computationally hard problem for the adversary, namely, finding the centralizer of a given finitely generated subgroup.

In this paper we address the question of representing the discriminant of an imaginary quadratic field with respect to the basis of a cyclotomic field. This representation allows us to parameterize new families of ordinary elliptic curves over finite prime fields suitable for pairing applications. In particular these curves have small discriminants greater than four and arbitrary embedding degree. Computational results are presented which support the theoretical findings.

In 1998, Blaze, Bleumer, and Strauss (BBS) proposed proxy re-signatures, in which a semi-trusted proxy acts as a translator between Alice and Bob. To translate, the proxy converts a signature from Alice into a signature from Bob on the same message. The proxy, however, does not learn any signing key and cannot sign arbitrary messages on behalf of either Alice or Bob. Since the BBS proposal, the proxy re-signature primitive has been largely ignored, but we show that it is a very useful tool for sharing web certificates, forming weak group signatures, and authenticating a network path. We begin our results by formalizing the definition of security for a proxy re-signature. We next substantiate the need for improved schemes by pointing out certain weaknesses of the original BBS proxy re-signature scheme which make it unfit for most practical applications. We then present two secure proxy re-signature schemes based on bilinear maps. Our first scheme relies on the Computational Diffie-Hellman (CDH) assumption; here the proxy can translate from Alice to Bob and vice-versa. Our second scheme relies on the CDH and 2-Discrete Logarithm (2-DL) assumptions and achieves a stronger security guarantee -- the proxy is only able to translate in one direction. Constructing such a scheme has been an open problem since proposed by BBS in 1998. Furthermore in this second scheme, even if the delegator and the proxy collude, they cannot sign on behalf of the delegatee. Both schemes are efficient and secure in the random oracle model.

We present a new signature whose security is reducible to a new assumptions about subgroups, the {\em Computational Conjugate Subgroup Members (CCSM) Assumption}, in the random oracle model.

In this paper, we suggest solutions to the key exposure problem in ring signature. In particular, we propose the first forward secure ring signature scheme and the first key-insulated ring signature schemes. Both constructions allow a $(t,n)$-threshold setting. That is, even $t$ secret keys are compromised, the validity of all forward secure ring signatures generated in the past is still preserved. In the other way, the compromise of up to all secret keys does not allow any adversary to generate a valid key-insulated ring signature for the remaining time periods.

Certificateless public-key cryptosystem is a recently proposed attractive paradigm using public key cryptosystem, which avoids the key escrow inherent in identity-based public-key cryptosystems, and does not need certificates to generate trust in public keys. In 2005, Al-Riyami and Paterson proposed a new certificateless public-key encryption scheme and proved its security in the random oracle model. This paper shows that their scheme is vulnerable to adaptive chosen ciphertext attacks, and presents a countermeasure to overcome such a security flaw.