Public Key Cryptography based on Semigroup Actions

A generalization of the original Diffie-Hellman key exchange in $(\Z/p\Z)^*$ found a new depth when Miller and Koblitz suggested that such a protocol could be used with the group over an elliptic curve. In this paper, we propose a further vast generalization where abelian semigroups act on finite sets. We define a Diffie-Hellman key exchange in this setting and we illustrate how to build interesting semigroup actions using finite (simple) semirings. The practicality of the proposed extensions rely on the orbit sizes of the semigroup actions and at this point it is an open question how to compute the sizes of these orbits in general and also if there exists a square root attack in general. In Section 2 a concrete practical semigroup action built from simple semirings is presented. It will require further research to analyse this system.Comment: 20 pages. To appear in Advances in Mathematics of Communication

Isogeny-based post-quantum key exchange protocols

The goal of this project is to understand and analyze the supersingular isogeny Diffie Hellman (SIDH), a post-quantum key exchange protocol which security lies on the isogeny-finding problem between supersingular elliptic curves. In order to do so, we first introduce the reader to cryptography focusing on key agreement protocols and motivate the rise of post-quantum cryptography as a necessity with the existence of the model of quantum computation. We review some of the known attacks on the SIDH and finally study some algorithmic aspects to understand how the protocol can be implemented

Public key exchange using semidirect product of (semi)groups

In this paper, we describe a brand new key exchange protocol based on a semidirect product of (semi)groups (more specifically, on extension of a (semi)group by automorphisms), and then focus on practical instances of this general idea. Our protocol can be based on any group, in particular on any non-commutative group. One of its special cases is the standard Diffie-Hellman protocol, which is based on a cyclic group. However, when our protocol is used with a non-commutative (semi)group, it acquires several useful features that make it compare favorably to the Diffie-Hellman protocol. Here we also suggest a particular non-commutative semigroup (of matrices) as the platform and show that security of the relevant protocol is based on a quite different assumption compared to that of the standard Diffie-Hellman protocol.Comment: 12 page

Exploring platform (semi)groups for non-commutative key-exchange protocols

In this work, my advisor Delaram Kahrobaei, our collaborator David Garber, and I explore polycyclic groups generated from number fields as platform for the AAG key-exchange protocol. This is done by implementing four different variations of the length-based attack, one of the major attacks for AAG, and submitting polycyclic groups to all four variations with a variety of tests. We note that this is the first time all four variations of the length-based attack are compared side by side. We conclude that high Hirsch length polycyclic groups generated from number fields are suitable for the AAG key-exchange protocol. Delaram Kahrobaei and I also carry out a similar strategy with the Heisenberg groups, testing them as platform for AAG with the length-based attack. We conclude that the Heisenberg groups, with the right parameters are resistant against the length-based attack. Another work in collaboration with Delaram Kahrobaei and Vladimir Shpilrain is to propose a new platform semigroup for the HKKS key-exchange protocol, that of matrices over a Galois field. We discuss the security of HKKS under this platform and advantages in computation cost. Our implementation of the HKKS key-exchange protocol with matrices over a Galois field yields fast run time

A Novel Provably Secure Key Agreement Protocol Based On Binary Matrices

In this paper, a new key agreement protocol is presented. The protocol uses exponentiation of matrices over GF(2) to establish the key agreement. Security analysis of the protocol shows that the shared secret key is indistinguishable from the random under Decisional Diffie-Hellman (DDH) assumption for subgroup of matrices over GF(2) with prime order, and furthermore, the analysis shows that, unlike many other exponentiation based protocols, security of the protocol goes beyond the level of security provided by (DDH) assumption and intractability of Discrete Logarithm Problem (DLP). Actually, security of the protocol completely transcends the reliance on the DLP in the sense that breaking the DLP does not mean breaking the protocol. Complexity of brute force attack on the protocol is equivalent to exhaustive search for the secret key