Group Theory in Lattice-Based Cryptography

Lecture Notes in Computer Science, 2015

A group signature allows a group member to anonymously sign messages on behalf of the group. In the past few years, new group signatures based on lattice problems have appeared: the most efficient lattice-based constructions are due to Laguillaumie et al. (Asiacrypt '13) and Langlois et al. (PKC '14). Both have at least O(n 2 log 2 n log N)bit group public key and O(n log 3 n log N)-bit signature, where n is the security parameter and N is the maximum number of group members. In this paper, we present a simpler lattice-based group signature, which is more efficient by a O(log N) factor in both the group public key and the signature size. We achieve this by using a new non-interactive zeroknowledge (NIZK) proof corresponding to a simple identity-encoding function. The security of our group signature can be reduced to the hardness of SIS and LWE in the random oracle model.

2018

Efficient member revocation and strong security against attacks are prominent requirements in group signature schemes. Among the revocation approaches Verifier-local revocation is the most flexible and efficient method since it requires to inform only the verifiers regarding the revoked members. The verifier-local revocation technique uses a token system to manage members’ status. However, the existing group signature schemes with verifier-local revocability rely on weaker security. On the other hand, existing static group signature schemes rely on a stronger security notion called, full-anonymity. Achieving the full-anonymity for group signature schemes with verifier-local revocation is a quite challenging task. This paper aims to obtain stronger security for the lattice-based group signature schemes with verifier-local revocability, which is closer to the full-anonymity. Moreover, this paper delivers a new key-generation method which outputs revocation tokens without derivin...

The gap factor γ of lattice problems (e.g., SVP, CVP, SIVP etc.) and sampling technique have an importance in lattice-based cryptography. In the ideal lattices, we introduce similar problems and expansion factor (like the gap factor in lattice problems). We also state some results related to ideal lattices.

Cryptogr., 2020

An efficient member revocation mechanism is a desirable feature when group signature schemes are applied in practical scenarios. Revocation methods, such as verifier-local revocation (VLR), provide an efficient member revocation in applications of group signatures. However, VLR-group signatures rely on a weaker security notion. On the other hand, group signature schemes for static groups gain stronger security with the full-anonymity security notion. Even though an outsider sees the secret signing keys of all group members in the full-anonymity, the signer is still anonymous. Achieving the full-anonymity for VLR group signature schemes is challenging due to the structure of secret signing keys. The secret signing keys of those schemes consist of tokens, which are used to manage revocation. The reveal of tokens may destroy the anonymity of the signers. We obtain stronger security for the lattice-based VLR group signature schemes by providing a new key generation method, which outputs...

Applicable Algebra in Engineering, Communication and Computing, 2006

Several suggestions are presented for developing cryptosystems, both classical and public key, using a combination of combinatorial group theory and linear groups. In particular, the Reidemeister-Schreier rewriting process is used as a one way function. These suggestions raise further questions concerning both implementation and security that are being explored in the thesis of Xu .

Designs, Codes and Cryptography, 2000

The public key cryptosystems M ST 1 and M ST 2 make use of certain kinds of factorizations of finite groups. We show that generalizing such factorizations to infinite groups allows a uniform description of several proposed cryptographic primitives. In particular, a generalization of M ST 2 can be regarded as a unifying framework for several suggested cryptosystems including the ElGamal public key system, a public key system based on braid groups and the MOR cryptosystem.

Computing Research Repository, 2011

Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The strength of computing machinery has made these techniques theoretically susceptible to attack and hence recently there has been an active line of research

math.gc.cuny.edu

Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The strength of computing machinery has made these techniques theoretically susceptible to attack and hence recently there has been an active line of research to develop cryptosystems and key exchange protocols using noncommutative cryptographic platforms. This line of investigation has been given the broad title of noncommutative algebraic cryptography. This was initiated by two public key protocols that used the braid groups, one by Ko, Lee et.al.and one by Anshel, Anshel and Goldfeld. The study of these protocols and the group theory surrounding them has had a large effect on research

In this Research investigation we présent new results in two areas-Cryptographic protocols and Lattice Problems. We introduce a new Protocolfor electronic cash which is exclusively designed to fonction on hardware with limited computing power. The approche has provable Security properties and low coputational requiremnts, but it still gives a fair amount of privacy. Major feature of the system is that there is no master secret that could be used for counterfeiting money if stolen. In this Research content we introduce the notion of hierarchial group signatures. In Merchant transactional process the signer that is as similar as a leaf of the sub tree of a group manager, the group manager learns which of its children that manages the signer. We introduced few practical conditions that are suitable for the new notion and construct a scheme that is provably secure given the existence of a family of trapdoor permutations. We also present a construction whcih is relatively practical, and prove its security in the rand om oracle model under the strong RSA assumtion and the DDH assumption. We déterminea simple and efficient system for electronic cash withprobable Security properties. The system relies on symmetric encryption technlogies rather than asymmetric.

International Journal of Research Publication and Reviews, 2023

The use of Mathematic in cryptography can result a safe encryption scheme. Lattices have emerged as a powerful mathematical tool in the field of cryptography, offering a diverse set of applications ranging from encryption to secure multi-party computation. This research paper provides a comprehensive review of the role of lattices in cryptography, covering both theoretical foundations and practical implementations. The paper begins by introducing the basic concepts of lattices and their relevance in cryptographic protocols. Subsequently, it explores key cryptographic primitives based on lattice problems, such as lattice-based encryption schemes, digital signatures, and fully homomorphic encryption. The paper also proposes a new lattice based cryptographic scheme.