On effective computations in subsemigroups of affine Cremona semigroup and implentations of new postquantum multivariate cryptosystems

Oleksandr Pustovit,Vasyl Ustimenko

doi:10.15407/fmmit2021.32.050

Abstract

Multivariate cryptography (MC) together with Latice Based, Hash based, Code based and Superelliptic curves based Cryptographies form list of the main directions of Post Quantum Cryptography.Investigations in the framework of tender of National Institute of Standardisation Technology (the USA) indicates that the potential of classical MC working with nonlinear maps of bounded degree and without the usage of compositions of nonlinear transformation is very restricted. Only special case of Rainbow like Unbalanced Oil and Vinegar digital signatures is remaining for further consideration. The remaining public keys for encryption procedure are not of multivariate. nature. The paper presents large semigroups and groups of transformations of finite affine space of dimension n with the multiple composition property. In these semigroups the composition of n transformations is computable in polynomial time. Constructions of such families are given together with effectively computed homomorphisms between members of the family. These algebraic platforms allow us to define protocols for several generators of subsemigroup of affine Cremona semigroups with several outputs. Security of these protocols rests on the complexity of the word decomposition problem, Finally presented algebraic protocols expanded to cryptosystems of El Gamal type which is not a public key system.

Highlights

Algebraic system on K[x1, x2,...xn ], where K is a commutative ring with operations of addition, multiplication and composition is the core part of Computer Algebra
The addition and multiplication of n polynomials from K[x1, x2,...xn ] of bounded degree can be computed in polynomial time but there is no polynomial algorithm for the execution of the computation of n elements from K[x1, x2,...xn ]
In 2017 the international tender of the National Institute of Standartisation Technology (NIST) of the USA for the selection of public key based on postquantum algorithms was announced

Summary

Introduction

Algebraic system on K[x1, x2,...xn ] , where K is a commutative ring with operations of addition, multiplication and composition is the core part of Computer Algebra. The addition and multiplication of n polynomials from K[x1, x2,...xn ] of bounded degree can be computed in polynomial time but there is no polynomial algorithm for the execution of the computation of n elements from K[x1, x2,...xn ]. It means that in Cremona semigroup СSn (K) (see [5]) of all endomorphisms of K[x1, x2,...xn ] the computation of the product of n. In 2017 the international tender of the National Institute of Standartisation Technology (NIST) of the USA for the selection of public key based on postquantum algorithms was announced It has been considering algorithms for the encryption task and for the procedure of digital signature.

Main results

Some implementations

Conclusions