On Multivariate Cryptosystems Based on Computable Maps with Invertible Decomposition

Let K be a commutative ring and K be an affine space over K of dimension n. We introduce the concept of a family of multivariate maps f(n) of K into itself with invertible decomposition. If f(n) is computable in polynomial time then it can be used as the public rule and the invertible decomposition provides a private key in f(n) based public key infrastructure. Requirements of polynomial ity of degree and density for f(n) allow to estimate the complexity of encryption procedure for a public user. The concepts of a stable family and a family of increasing order are motivated by the studies of discrete logarithm problem in the Cremona group. The statement on the existence of families of multivariate maps of polynomial degree and polynomial density of increasing order with the invertible decomposition is proved. The proof is supported by explicite construction which can be used as a new cryptosystem. The presented multivariate encryption maps are induced by special walks in the algebraically defined extremal graphs A(n,K) and D(n,K) of increasing girth. 1 On Multivariate Cryptography and Special Multivariate Transformations Multivariate cryptography (see [1]) is one of the directions of Postquantum Cryptography, which concerns algorithms resistant to hypothetic attacks conducted by Quantum Computer. The encryption tools of Multivariate Cryptography are nonlinear multivariate transformations of affine space K, where K is a finite commutative ring. Nowadays this modern direction of research requires new examples of algorithms with theoretical arguments on their resistance to attacks conducted by ordinary computer (Turing machine) and new tasks for cryptanalists. ∗[email protected] Pobrane z czasopisma Annales AIInformatica http://ai.annales.umcs.pl Data: 19/01/2018 02:15:27