Abstract
Abstract We offer a public key exchange protocol based on a semidirect product of two cyclic (semi)groups of matrices over Z p {{\mathbb{Z}}}_{p} . One of the (semi)groups is additive, and the other one is multiplicative. This allows us to take advantage of both operations on matrices to diffuse information. We note that in our protocol, no power of any matrix or of any element of Z p {{\mathbb{Z}}}_{p} is ever exposed, so standard classical attacks on Diffie–Hellman-like protocols are not applicable.
Highlights
We start by recalling the classical Diffie–Hellman protocol [1]
A more general description of the protocol uses an arbitrary finite cyclic group: (1) Alice and Bob agree on a finite cyclic group G of order q and a generating element g in G
Since mn = nm, both Alice and Bob are in possession of the same group element K = KA = KB, which can serve as the shared secret key
Summary
We start by recalling the classical Diffie–Hellman protocol [1]. The simplest, and original, implementation of this protocol uses the multiplicative group of integers modulo p, where p is prime and g is primitive modulo p. Since mn = nm, both Alice and Bob are in possession of the same group element K = KA = KB, which can serve as the shared secret key. The eavesdropper must solve the Diffie–Hellman problem (recover gmn from g, gm, and gn) to obtain the shared secret key. This is currently considered difficult for a “good” choice of parameters [3], a new key exchange protocol was offered, based on a semidirect product of multiplicative matrix semigroups. That protocol is similar to the Diffie–Hellman protocol, but it differs in one essential detail: at the last two steps, Alice and Bob use multiplication instead of exponentiation. We note that in our protocol, no power of any matrix is ever exposed, so standard classical attacks on Diffie–Hellman-like protocols are not applicable
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