In this paper, we do a cryptanalyse of the so called "Strong Diffie-Hellman-DSA Key Exchange (briefly: SDH-DSA-KE)" and after we propose "Strong Diffie-Hellman-Exponential-Schnnor Key Exchange (briefly: SDH-XS-KE)" which is an improvement for efficiency and security. SDH-XS-KE protocol is secure against Session State Reveal (SSR) attacks, Key independency attacks, Unknown-key share (UKS) attacks and Key-Compromise Impersonation (KCI) attacks. Furthermore, SDH-XS-KE has Perfect Forward Secrecy (PFS) property and a key confirmation step. The new proposition is not vulnerable to Disclosure to ephemeral or long-term Diffie-Hellman exponents. We design our protocol in finite groups therefore this protocol can be implemented in elliptic curves.

NTRU is the first public key Cryptosystem based on the polynomial ring Z[X] X N −1. The hard problem underlying this cryptosystem is related to finding short vectors in a lattice. Several generalizations of NTRU was designed over various integral ring such as Z, Z[i], Z[w] and H. In this paper, we use the ring with zeros divisors D = Z + Z, 2 = 0 (called the ring of Dual Integers) in order to design a new version of NTRU. To achieve this objective, we have studied the elementary arithmetic properties of the ring of Dual integers in a previous paper. The main difficulty is to be able to perform a division algorithm with a unique remainder and to invert polynomials with coefficients in quotient ring of the ring of Dual integers. Nevertheless, we have successfully design NTRU over Dual integers (called DTRU) in a particular quotient ring of the ring of Dual integers. Our scheme has the same level security than NTRU, but is not more efficient. This work shows also that NTRU can be designed even if the ring has zeros divisors! We have also design over the ring of Dual Integer the cryptosystem NTRU with Non-inverible polynomial proposed by Banks and Shparlinski. This this version is more secure than NTRU but is less efficient too.

This paper introduces the scalar multiplication on Huff elliptic curves defined over a finite field of even characteristic using the Frobenius expansion.

In our paper paper we propose a new binary elliptic curve of the form $a[x^2+y^2+xy+1]+(a+b)[x^2y+y^2x]=0$. If $mgeq 5$ we prove that each ordinary elliptic curve $y^{2}+xy=x^{3}+alpha x^2+eta,  etaeq 0$ over $mathbb{F}_{2^m}$, is birationally equivalent over $mathbb{F}_{2^m}$ to our curve. This paper also presents the formulas for the group law

This paper introduces the Frobenius endomorphism on the the binary Edwards elliptic curves proposed by Bernstein, Lange and Farashahi in 2008 and by Diao and Lubicz (2010). To speed up the scalar multiplication on binary Edwards curves, we use the GLV method combined with the Frobenius endomorphism over the curve.