Алгоритмы ЭЦП на конечных некоммутативных алгебрах над полями характеристики два

Abstract

Abstract Purpose of work is the increase in performance and decrease in the hardware implementation cost of the of post-quantum algebraic signature algorithms based on the computational difficulty of solving systems of many quadratic equations with many unknowns. Research method is i) the development of post-quantum signature algorithms on finite non-commutative associative algebras defined over finite fields of characteristic two, which have high performance and small sizes of signature and public and secret keys; ii) using the concept of constructing algebraic signature algorithms with a hidden commutative group, characterized by the use of a power-type vector verification equation with multiple occurrences of the signature S as a factor; iii) the choice of the degree of extension z of the field GF(2z) in which the order of the hidden group is divisible only by prime divisors of at least 24 bits. Results of the study are the formulated main provisions for the implementation of post-quantum signature algorithms with a hidden group, the security of which is based on the computational difficulty of solving systems of many quadratic equations with many unknowns, when using the finite non-commutative algebras given over the GF(2z) fields as algebraic support. The values of the extension degree z are established for which the order of the hidden commutative group is divisible only by prime divisors of a sufficiently large size. A new post-quantum signature algorithm with relatively high performance and small sizes of the signature and public and secret keys have been developed. Using an informal security index in the form of a product of the binary logarithm of the order of the field and the number of unknowns, the developed and known post-quantum algorithms for a given level of security are compared. Practical relevance. The main provisions for constructing signature algorithms with a hidden group are formulated for the case of using finite non-commutative algebras with computationally efficient operations of multiplication and exponentiation, providing prerequisites for improving performance and reducing the hardware implementation cost of post-quantum signature algorithms.