Алгебраические алгоритмы ЭЦП, основанные на трудности решения систем уравнений

Abstract

Purpose of work is the development of post-quantum digital signature algorithms with comparatively small sizes of the public and secret keys and the signature. Research method is the use of a new concept for constructing signature algorithms on finite non-commutative associative algebras, which is distinguished by the multiple occurrences of the signature S in the power verification equation. A public key is generated in the form of a set of vectors every of which is calculated as the product of triples of secret vectors. With a special choice of these triples, it is possible to calculate a signature that satisfies the verification equation. Results of the study are two developed algebraic post-quantum digital signature algorithms of a new type, security of which is based on the computational difficulty of solving systems of many quadratic equations with many unknowns. The difference from the public-key algorithms of multivariate cryptography is that the system of quadratic equations is derived from the formulas for generating the public-key elements in the form of a set of vectors of m-dimensional finite non-commutative algebra with an associative vector multiplication operation. The said formulas define the system of n quadratic vector equations, which reduces to the system of mn quadratic equations over a finite field. Thanks to the “natural” mechanism for the occurrence of the specified system, it is set above the field, the order of which has a large size (97 and 129 bits). The used procedures for generating the public key and signature include the exponentiation operations to the degree of a large size (96 and 128 bits), which are performed over the elements of the secret (hidden) commutative group contained in the algebra. The signature is formed in the form of two elements: a randomizing natural number e and a “fitting” vector S. The signature authentication equation includes a multiple occurrence of the S element and every entry of the vector S is associated with the formation of a product that is exponentiated to a degree dependent on the value of the e element. A significant reduction in the size of public and secret keys and signatures has been achieved, as well as an increase in performance compared to foreign analogues, considered currently as basic algorithms for the adoption of post-quantum digital signature standards.