Constructions in public-key cryptography over matrix groups

ISBN : 978-0-8218-4037-5International audienceThe purpose of the paper is to give new key agreement protocols (a multi-party extension of the protocol due to Anshel-Anshel-Goldfeld and a generalization of the Diffie-Hellman protocol from abelian to solvable groups) and a new homomorphic public-key cryptosystem. They rely on difficulty of the conjugacy and membership problems for subgroups of a given group. To support these and other known cryptographic schemes we present a general technique to produce a family of instances being matrix groups (over finite commutative rings) which play a role for these schemes similar to the groups $Z_n^*$ in the existing cryptographic constructions like RSA or discrete logarithm

Authentication from matrix conjugation

We propose an authentication scheme where forgery (a.k.a. impersonation) seems infeasible without finding the prover's long-term private key. The latter would follow from solving the conjugacy search problem in the platform (noncommutative) semigroup, i.e., to recovering X from X^{-1}AX and A. The platform semigroup that we suggest here is the semigroup of nxn matrices over truncated multivariable polynomials over a ring.Comment: 6 page

Post-Quantum Cryptography: S

Currently there is an active Post-Quantum Cryptography (PQC) solutions search, which attempts to find cryptographic protocols resistant to attacks by means of for instance Shor's polynomial time algorithm for numerical field problems like integer factorization (IFP) or the discrete logarithm (DLP). The use of non-commutative or non-associative structures are, among others, valid choices for these kinds of protocols. In our case, we focus on a permutation subgroup of high order and belonging to the symmetric group S381. Using adequate one-way functions (OWF), we derived a Diffie-Hellman key exchange and an ElGamal ciphering procedure that only relies on combinatorial operations. Both OWF pose hard search problems which are assumed as not belonging to BQP time-complexity class. Obvious advantages of present protocols are their conceptual simplicity, fast throughput implementations, high cryptanalytic security and no need for arithmetic operations and therefore extended precision libraries. Such features make them suitable for low performance and low power consumption platforms like smart cards, USB-keys and cellphones

The tropical version of El Gamal Encryption

In this paper, we consider the new version of tropical cryptography protocol, i.e the tropical version of El Gamal encryption.  We follow the ideas and modify the clasical El Gamal encryption using tropical matrices and matrix power in tropical algebra. Then we also provide a toy example for the reader’s understanding.

Tropical cryptography II: extensions by homomorphisms

We use extensions of tropical algebras as platforms for very efficient public key exchange protocols.Comment: 7 pages. arXiv admin note: text overlap with arXiv:1301.119