A NEW CONCEPT FOR DESIGNING POST-QUANTUM DIGITAL SIGNATURE ALGORITHMS ON NON-COMMUTATIVE ALGEBRAS
Authors:   St. Petersburg Federal Research Center of the Russian Academy of Sciences, Moldovyan Dmitriy , Moldovyan Alexander ,   St. Petersburg Federal Research Center of the Russian Academy of Sciences, Moldovyan Nikolay ,   St. Petersburg Federal Research Center of the Russian Academy of Sciences
Published: January 2022

Purpose of work is the development of a new approach to designing post-quantum digital signature algorithms that are free from the shortcomings of known analogs – large sizes of the signature and public key. Research method is the use of power vector equations with multiple occurrences of the signature S as a signature verification equation. The computational difficulty of solving equations of the said type relatively the unknown value of S ensures the resistance of the signature scheme to attacks using S as a fitting parameter. The possibility of calculating the value of S by the secret key is provided by using the public key in the form of a set of secret elements of the hidden group, masked by performing left and right multiplications by matched invertible vectors. Results of the study include a new proposed concept for the development of post-quantum digital signature algorithms on non-commutative algebras, which use a hidden commutative group. One of its main differences is the use of a secret key in the form of a set of vectors, the knowledge of which makes it possible to calculate the correct signature value for the random powers present in the verification equation. The form of the latter defines a system of quadratic vector equations connecting the public key with the secret, which is reduced to a system of many quadratic equations with many unknowns, given over a finite field. The computational difficulty of finding a solution to the latter system determines the security of the algorithms developed within the framework of the proposed concept. A quantum computer is ineffective for solving this problem, therefore, the said algorithms are post-quantum. As analogs in construction, digital signature algorithms based on the computational difficulty of the hidden discrete logarithm problem are considered, however, the use of a hidden group and exponentiation operations represent only a general technique for ensuring the correctness of the signature schemes developed within the framework of the concept, and not for specifying a basic computationally difficult problem. To improve the performance of the signature generation and verifications procedures, the four-dimensional algebras defined by sparse basis vector multiplication tables are used as an algebraic support. The proposed concept is confirmed by the development of a specific post-quantum algorithm that provides a significant reduction in the size of the public key and signature in comparison with the finalists of the NIST global competition in the nomination of post-quantum digital signature algorithms. Practical relevance: The developed new concept for constructing post-quantum digital signature algorithms expands the areas of their application in conditions of limited computing resources