A Novel Method for Developing Post-quantum Digital Signature Algorithms on Non-commutative Associative Algebras

Abstract

Introduction: Development of practical post-quantum signature algorithms is a current challenge in the area of cryptography. Recently, several candidates on post-quantum signature schemes, in which the exponentiation operations in a hidden commutative group contained in a non-commutative algebra is used, were proposed. Search for new mechanisms of using a hidden group, while developing signature schemes resistant to quantum attacks, is of significant practical interest. Purpose: Development of a new method for designing post-quantum signature algorithms on finite non-commutative associative algebras. Results: A novel method for developing digital signature algorithms on non-commutative algebras. A new four-dimensional finite non-commutative associative algebra set over the ground field GF(p) have been proposed as algebraic support of the signature algorithms. To provide a higher performance of the algorithm, in the introduced algebra the vector multiplication is defined by a sparse basis vector multiplication table. Study of the algebra structure has shown that it can be represented as a set of commutative subalgebras of three different types, which intersect exactly in the set of scalar vectors. Using the proposed method and introduced algebra, a new post-quantum signature scheme has been designed. The introduced method is characterized in using one of the elements of the signature (e, S) in form of the four-dimensional vector S that is computed as a masked product of two exponentiated elements G and H of a hidden commutative group: S = B-1GnHmC-1, where non-permutable vectors B and C are masking multipliers; the natural numbers n and m are calculated depending on the signed document M and public key. The pair <G, H> composes a minimum generator systems of the hidden group. The signature verification equation has the form R = (Y1SZ1)e(Y2SZ2)e2, where pairwise non-permutable vectors Y1, Z1, Y2, and Z2 are element of the public key and natural number e that is computed depending on the value M and the vector R. Practical relevance: Due to sufficiently small size of public key and signature and high, the developed digital signature scheme represents interest as a practical post-quantum signature algorithm. The introduced method is very attractive to develop a post-quantum digital signature standard.

Signature Algorithms with a Hidden Group, Based on Difficulty of Solving Systems of Quadratic Equations

“…However, there are problems justifying their post-quantum security associated with the potential possibility of reducing the hidden DLP to the usual DLP. At the beginning of 2022, another type of hidden-group algebraic algorithm was proposed, characterized by the fact that it uses the computational difficulty of finding a solution to a system of many quadratic equations with many unknowns [15]. Taking into account that quantum computers are not efficient for solving the latter problem [16,17], we can say that the second type of algebraic algorithm with a hidden group is of special interest for developing practical post-quantum signature algorithms.…”

“…A high assessment of the results obtained in the MPKC research area is the selection of the Rainbow algorithm as one of the finalists and of the GeMSS as one of the alternative algorithms in the NIST competition [11] in the nomination of post-quantum digital signature algorithms. The MPKC algorithms represented the single type of signature schemes based on the computational difficulty of solving a system of large square equations until 2022 when an alternative method for constructing algorithms based on the said problem was proposed in [15]. That paper introduced a novel method for designing digital signature algorithms with a hidden group, using Finite Non-commutative Associative Algebras (FNAA) as algebraic support, and proposed a specific algorithm implementing that method, using a verification equation with two entries of the signature.…”

“…That paper introduced a novel method for designing digital signature algorithms with a hidden group, using Finite Non-commutative Associative Algebras (FNAA) as algebraic support, and proposed a specific algorithm implementing that method, using a verification equation with two entries of the signature. A significant merit of the method [15] is the small size of both the signature and the public key, which provides the potential possibility of developing practical post-quantum signature algorithms and a post-quantum signature standard.…”

“…However, in Ref. [15], the issues of justification of the selection of the dimension of the FNAA used as algebraic support and the choice of the 256-bit order of the finite field, over which the FNAA is set, had not been considered.…”

“…The design of the algorithms allows using the FNAAs set over the ground fields GF(p), with the order p having a size of 81 to 129 bits. The latter provides a smaller public key and signature in comparison with the signature algorithm from [15] at the same security level.…”