**On semigroups of multivariate transformations constructed in terms of time dependent linguistic graphs and solutions of Post Quantum Multivariate Cryptography.**

*V. Ustimenko *

**Abstract: **Time dependent linguistic graphs over abelian group H are introduced. In the case $H=K*$ such bipartite graph with point set $P=H^n$ can be used for generation of Eulerian transformation of $(K*)^n$, i.e. the endomorphism of $K[x_1, x_2,… , x_n]$ sending each variable to a monomial term. Subsemigroups of such endomorphisms together with their special homomorphic images are used as platforms of cryptographic protocols of noncommutative cryptography.
The security of these protocol is evaluated via complexity of hard problem of decomposition of Eulerian transformation into the product of known generators of the semigroup. Nowadays the problem is intractable one in the Postquantum setting.
The symbiotic combination of such protocols with special graph based stream ciphers working with plaintext space of kind $K^m$ where $m=n^t$ for arbitrarily chosen parameter $t$ is proposed.
This way we obtained a cryptosystem with encryption/decryption procedure of complexity $O(m^{1+2/t})$.

**Category / Keywords: **cryptographic protocols / Post Quantum Cryptography, Computer Algebra, time dependent algebraic graphs, affine Cremona semigroup, Eulerian transformations, linguistic graphs over groups and commutative rings.

**Date: **received 2 Nov 2021

**Contact author: **vasyl at hektor umcs lublin pl

**Available format(s): **PDF | BibTeX Citation

**Note: **We present protocol based cryptosystem which is not a public key algorithm.

**Version: **20211106:154915 (All versions of this report)

**Short URL: **ia.cr/2021/1466

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