**Entropoid Based Cryptography**

*Danilo Gligoroski*

**Abstract: **The algebraic structures that are non-commutative and non-associative known as entropic groupoids that satisfy the "Palintropic" property i.e., $x^{\mathbf{A} \mathbf{B}} = (x^{\mathbf{A}})^{\mathbf{B}} = (x^{\mathbf{B}})^{\mathbf{A}} = x^{\mathbf{B} \mathbf{A}}$ were proposed by Etherington in '40s from the 20th century. Those relations are exactly the Diffie-Hellman key exchange protocol relations used with groups. The arithmetic for non-associative power indices known as Logarithmetic was also proposed by Etherington and later developed by others in the 50s-70s. However, as far as we know, no one has ever proposed a succinct notation for exponentially large non-associative power indices that will have the property of fast exponentiation similarly as the fast exponentiation is achieved with ordinary arithmetic via the consecutive rising to the powers of two.
In this paper, we define ringoid algebraic structures $(G, \boxplus, *)$ where $(G, \boxplus) $ is an Abelian group and $(G, *)$ is a non-commutative and non-associative groupoid with an entropic and palintropic subgroupoid which is a quasigroup, and we name those structures as Entropoids. We further define succinct notation for non-associative bracketing patterns and propose algorithms for fast exponentiation with those patterns.
Next, by an analogy with the developed cryptographic theory of discrete logarithm problems, we define several hard problems in Entropoid based cryptography, such as Discrete Entropoid Logarithm Problem (DELP), Computational Entropoid Diffie-Hellman problem (CEDHP), and Decisional Entropoid Diffie-Hellman Problem (DEDHP). We post a conjecture that DEDHP is hard in Sylow $q$-subquasigroups. Next, we instantiate an entropoid Diffie-Hellman key exchange protocol. Due to the non-commutativity and non-associativity, the entropoid based cryptographic primitives are supposed to be resistant to quantum algorithms. At the same time, due to the proposed succinct notation for the power indices, the communication overhead in the entropoid based Diffie-Hellman key exchange is very low: for 128 bits of security, 64 bytes in total are communicated in both directions, and for 256 bits of security, 128 bytes in total are communicated in both directions.
Our final contribution is in proposing two entropoid based digital signature schemes. The schemes are constructed with the Fiat-Shamir transformation of an identification scheme which security relies on a new hardness assumption: computing roots in finite entropoids is hard. If this assumption withstands the time's test, the first proposed signature scheme has excellent properties: for the classical security levels between 128 and 256 bits, the public and private key sizes are between 32 and 64, and the signature sizes are between 64 and 128 bytes. The second signature scheme reduces the finding of the roots in finite entropoids to computing discrete entropoid logarithms. In our opinion, this is a safer but more conservative design, and it pays the price in doubling the key sizes and the signature sizes.
We give a proof-of-concept implementation in SageMath 9.2 for all proposed algorithms and schemes in an appendix.

**Category / Keywords: **public-key cryptography / Post-quantum cryptography, Discrete Logarithm Problem, Diffie-Hellman key exchange, entropic, Entropoid, Entropoid Based Cryptography

**Date: **received 12 Apr 2021

**Contact author: **danilog at ntnu no

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

**Version: **20210412:180250 (All versions of this report)

**Short URL: **ia.cr/2021/469

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