Non-associative public-key cryptography

Abstract

We introduce a generalized Anshel-Anshel-Goldfeld (AAG) key establishment protocol (KEP) for magmas. This leads to the foundation of non-associative public-key cryptography (PKC), generalizing the concept of non-commutative PKC. We show that left selfdistributive systems appear in a natural special case of a generalized AAG-KEP for magmas, and we propose, among others instances, concrete realizations using fconjugacy in groups and shifted conjugacy in braid groups. We discuss the advantages of our schemes compared with the classical AAG-KEP based on conjugacy in braid groups.KEP's based on the shifted and f -conjugacy problem. In section 5 we discuss generalizations, like AAG-schemes over non-associative magmas, open problems and further work.Summary. The main purpose of this paper is popularize the notion of nonassociative cryptography and to provide a general framework for non-associative and non-commutative KEP's by utilizing the unifying approach that stems from the general AAG-KEP for magmas. We argue for the superiority of the non-associative schemes introduced in section 4 compared to classical noncommutative AAG commutator KEP. Anyway, in our opinion the field of non-commutative cryptography lacked over the last years supply of new innovative cryptosystems. We hope that nonassociative cryptography will contribute to revived interest in non-commutative cryptography.Outlook. Nevertheless, this is not the end, rather the beginning of the story of non-associative cryptography.In the forthcoming paper [KaT12], by introducing a small asymmetry in the nonassociative AAG protocol for magmas, we succeed to construct non-associative KEP's for all LD-and multi-LD-systems (in general: sets with distributive operations). We consider the systems and instances given in [KaT12] as much more practical and interesting than the one given in this paper. In particular, since these systems work for all LD-and multi-LD-systems, they deploy two further advantages. First, we may consider encryption functions using iterated multiplication (in the magma) from the left. Therefore, in order to obtain the secret key an attacker has to solve an iterated f -or shifted conjugacy problem. Second, for a given (partial) multi-LD-system it turns out that even the used operations can be hidden, i.e., they are part of the secret key.Historical remarks. Non-associative structures, in particular quasigroups seem to have a long history in cryptography. For an overview on cryptographic applications of quasigroups and Latin squares, see [Shc09, GS10, Shc12]. In particular, we mention the work of Denes and Keedwell [DK74, DK91, DK92, DK02]. Nevertheless, except for authentication schemes and zero-knowledge protocols, most of these applications are in classical (i.e. symmetric key) cryptography. The earliest quasigroup-based public-key cryptosystem that we are aware of is due to Koscielny and Mullen [KM99]. Non-associative cryptography that goes beyond quasigroups, in particular, the generalized AAG-KEP for magmas were introduced by the author in his...

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