On the usage of postquantum protocols defined in terms of transformation semigroups and their homomophisms

Abstract

We suggest new applications of protocols of Non-commutative cryptography defined in terms of subsemigroups of Affine Cremona Semigroups over finite commutative rings and their homomorphic images to the constructions of possible instruments of Post Quantum Cryptography. This approach allows to define cryptosystems which are not public keys. When extended protocol is finished correspondents have the collision multivariate transformation on affine space Kn or variety (K*)n where K is a finite commutative ring and K* is nontrivial multiplicative subgroup of K. The security of such protocol rests on the complexity of word problem to decompose element of Affine Cremona Semigroup given in its standard form into composition of given generators. The collision map can serve for the safe delivery of several bijective multivariate maps Fi (generators) on Kn from one correspondent to another. So asymmetric cryptosystem with nonpublic multivariate generators where one side (Alice) knows inverses of Fi but other does not have such a knowledge is possible. We consider the usage of single protocol or combinations of two protocols with platforms of different nature. The usage of two protocols with the collision spaces Kn and (K*)n allows safe delivery of two sets of generators of different nature. In terms of such sets we define an asymmetric encryption scheme with the plainspace (K*)n, cipherspace Kn and multivariate non-bijective encryption map of unbounded degree O(n) and polynomial density on Kn with injective restriction on (K*)n. Algebraic cryptanalysis faces the problem to interpolate a natural decryption transformation which is not a map of polynomial density