Categorical interpretations of some key agreement protocols

Key agreement is one of the fundamental cryptographic primitives after encryption and digital signature. Key agreement protocols (KAPs) allow two or more parties to exchange information among themselves over an adversarially controlled insecure network and agree upon a common session key, which may be used for later secure communication among the parties. Thus, secure KAPs serve as basic building block for constructing secure, complex, higher level cryptographic protocols.

The first pioneering work for key agreement is the Diffie-Hellman protocol given in their seminal paper [2] that invents the public key cryptography and revolutionizes the field of modern cryptography. In [2] a two-party key agreement protocol was proposed. There have been many attempts to provide authentic key agreement based on the Diffie-Hellman protocol [3,6,7,9].

In the last few years some efforts have been made to construct KAP using hard problems in infinite non-commutative groups. Here we only mention the idea based on conjugacy search problem which were reckoned as potentially hard problem for construction of one-way functions [1,4]. To realize proposed algorithms the main attempts were directed to the suitable platform group selection.

Recently in [8] the KAP has been constructed using matrix power functions based on matrix ring action on some matrix set and generalizing the Diffie-Hellman KAP. It has been suggested that main advantage of the proposed KAP is considerable fast computations and avoidance of arithmetic operations with long integers.

The aim of this work is to suggest a general scheme of constructing KAPs using the category theory. We assume the reader is familiar with categories (we refer to the classical book of Mac Lane [5] for the background in Category Theory). Based on the structure of categories, we present the above mentioned KAPs as very particular cases of our categorical KAPs. Working new examples of our categorical KAPs will be given in subsequent papers.

In this section we define KAPs which are arisen from the structure of categories.

2.1. KAP based on categories. Let C be a (non-empty) category and let A, B be objects of C such that Hom(A, B) = ∅. We suggest the set Hom(A, B) to be a set of possible keys, while Hom(A, A) and Hom(B, B) are monoids which can be used by Alice and Bob, respectively, for actions on Hom(A, B) if they wish to create a shared key. According to the structure of the category C, Alice is able to act on the set of possible keys using the right action of Hom(A, A) on Hom(A, B). Similarly, Bob is able to act on the set of possible keys using the left action of Hom(B, B) on Hom(A, B). Let g be a publicly known element of the set Hom(A, B). Then, for creating a shared key, Alice and Bob can proceed as follows:

1. Alice selects at random an element f ∈ Hom(A, A) and computes composition g • f , and sends it to Bob; 2. Bob selects at random an element h ∈ Hom(B, B) and computes composition h • g, and sends it to Alice; 3. Alice computes

This protocol, based on the structure of the category C, is called the categorical key agreement protocol (CKAP).

In this subsection we give another scheme of KAP induced by a structure of a category, but which is enriched over the category of abelian groups, i.e. a category whose morphism sets are abelian groups satisfying some axioms (see [5]). This construction generalizes the KAP given in previous subsection and motivated by some known KAPs. Namely, our approach makes it possible to interpret many known KAPs as particular cases of our construction.

Let D be a (non-empty) enriched category over the category of abelian groups. Clearly, it means that for any objects A and B in this category Hom(A, A) and Hom(B, B) are unital rings, Hom(A, B) is an abelian group and composition of morphisms in D is bilinear. Let A, B be objects of D such that Hom(A, B) = ∅. Let m, n ∈ N be natural numbers, A A and B A commuting subrings of the n × n-matrix ring M n Hom(A, A) , while A B and B B commuting subrings of m × m-matrix ring M m Hom(B, B) . Let ϕ be a publicly known m × n-matrix over the abelian group Hom(A, B). If Alice and Bob wish to create a common secret key, they can proceed as follows:

This protocol is called the enriched categorical key agreement protocol (ECKAP). The following assertion relates two categorical KAPs presented in this section.

Theorem 2.1. There is a universal faithful functor T from the category of categories to the category of enriched categories over the category of abelian groups. According to this correspondence, any CKAP related to a category C can be interpreted as a ECKAP related to the enriched category T (C).

Proof. We just construct the functor T and omit the proof of its universality since it directly follows from the construction. In fact, for any category C define the category T (C) as follows: its objects class coincides with the objects class of C, while Hom T (C) (A, B) is the free abelian group generated by

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