Post-quantum signature algorithms on noncommutative algebras, using difficulty of solving systems of quadratic equations

Abstract

A recently proposed new concept for constructing algebraic signature schemes with a hidden group is used to develop two new post-quantum signature algorithms on four-dimensional and six-dimensional finite noncommutative associative algebras. As in the case of the post-quantum signature schemes of multivariate cryptography, the security of the introduced algorithms is based on the computational difficulty of solving systems of many quadratic equations (44 and 42) with many unknowns (40 and 36). The signature represents a pair of a natural number e and a vector S. The latter enters three times in the verification equation used, providing resistance to the forging signature attacks by using the value S as a fitting parameter. A public key is generated in the form of a set of vectors, each of which is calculated as the product of triples of secret vectors. With a special choice of these triples, a signer has the possibility of calculating a signature that satisfies the verification equation. The developed post-quantum signature algorithms are practical, having sufficiently small signatures (97 and 109 bytes), public keys (387 and 291 bytes), and secret keys (315 and 451 bytes). A significant difference from the public-key algorithms of multivariate cryptography is that in the developed signature schemes, 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 noncommutative algebra with an associative vector multiplication operation. The formulas define the system of n quadratic vector equations, which reduces to the system of m quadratic equations over a finite field.