Signature Algorithms on Non-commutative Algebras Over Finite Fields of Characteristic Two

Abstract

The paper considers digital signature algorithms with a hidden group, security of which is based on the computational difficulty of solving a system of many quadratic equations with many unknowns. The attention is paid to an implementation of the said type of algorithms on finite non-commutative associative algebras set over the finite fields of characteristics two. The use of the latter type of algebraic support is aimed to improving the performance and reducing the hardware implementation cost. A new algebraic algorithm with a hidden group is introduced, in which a four-dimensional non-commutative algebra is used as algebraic support. In the used algebra the vector multiplication operation is defined by a sparse basis vector multiplication table. Decomposition of the non-commutative algebra into set of commutative subalgebras is studied. The formulas describing the number of the subalgebras of every type are also presented. It is shown that the factorization of the order of the hidden group is non-critical for the security of the signature algorithm, so one can apply the GF(2z) fields with a sufficiently large number of different values of the degree z, including those that are equal to a Mersenne exponent.