Digital signature scheme on the 2 x 2 matrix algebra

Abstract

The article considers the structure of the 2x2 matrix algebra set over a ground finite field GF(p). It is shown that this algebra contains three types of commutative subalgebras of order p2, which differ in the value of the order of their multiplicative group. Formulas describing the number of subalgebras of every type are derived. A new post-quantum digital signature scheme is introduced based on a novel form of the hidden discrete logarithm problem. The scheme is characterized in using scalar multiplication as an additional operation masking the hidden cyclic group in which the basic exponentiation operation is performed when generating the public key. The advantage of the developed signature scheme is the comparatively high performance of the signature generation and verification algorithms as well as the possibility to implement a blind signature protocol on its base.

A New Method for Developing Signature Algorithms on Finite Non-commutative Algebras

“…From the point of view of the decomposition into a set of commutative subalgebras the FNAAs set by Tables 2 and 3 have a similar structure. Each of these algebras is divided into commutative subalgebras of order p 2 , which are attributed exactly to the following three types [10,14]: i) subalgebras of Ψ 1 type include a multiplicative group of the order (p − 1) 2 , which has 2-dimensional cyclicity;…”

“…The Ψ subalgebras intersect exactly in the set of scalar vectors. The number η of the Ψ 1 , Ψ 2 , and Ψ 3 subalgebras are equal respectively to (see [10,14]):…”

“…In this paper we develop a post-quantum signature scheme using a 4-dimensional FNAA as an algebraic support, in which the vector multiplication operation is set over GF (p), where p = 2q + 1 and q is a 256-bit prime, by a sparse BVMT (in order to get a higher performance of the post-quantum signature scheme). Table 2 [10] and Table 3 [14] suit well as an algebraic support of the developed algorithm. Note that in the case of Table 3 and structural coefficient λ = 1 we have 2x2 matrix algebra.…”

“…The first and the second occurrence is connected with computation of the vectors Y 1 SZ 1 and Y 2 SZ 2 which are exponentiated to different powers depending on a natural number that represents the randomization element of the signature (e, S). Thus, procedure for generating a signature to an electronic Table 3 Setting the 4-dimensional FNAA with the two-sided unit (1, 0, 0, 1) [14]; λ = 0.…”

“…The interest in the HDLP problem is related to the fact that the HDLP-based signature schemes have relatively small sizes of the public key and signature. This area of research is quite new, and for a deeper and more complete understanding of the possibilities for the development of practical post-quantum HDLP-based signature schemes, it is of significant interest to search for new forms of the HDLP, especially for the case of using 4-dimensional FNAAs as a carrier of the HDLP, which are defined using sparse basis vector multiplication tables (BVMT) [10,14]. The main feature of the HDLP-based signature schemes is the use of a hidden group in which the exponentiation operations are executed, while generating the public key and generating and verifying signatures.…”