In this paper the family of elliptic curves over Q given by the equation Ep :Y2 = (X - p)3 + X3 + (X + p)3 where p is a prime number, is studied. Itis shown that the maximal rank of the elliptic curves is at most 3 and someconditions under which we have rank(Ep(Q)) = 0 or rank(Ep(Q)) = 1 orrank(Ep(Q))≥2 are given.

The possibility of using elliptic curves over the rational field non-zero ranks in cryptographic schemes is studied. For first time, construction cryptosystems proposed security which based on complexity solving knapsack problem numbers ranks. A new approach to use for proposed. few experiments have been carried out estimate heights characteristic points infinite order. model a cryptosystem res...

In this paper, we build, in a generic way, two asymmetric cryptosystems with a careful study of their security. We present first an additively homomorphic scheme which generalizes, among others, the Paillier cryptosystem, and then, another scheme, built from a deterministic trapdoor function. Both schemes are proved semantically secure against chosen plaintext attacks in the standard security m...

Addition chain calculations play a critical role in determining the e ciency of cryptosystems based on isogenies on elliptic curves. However, nding a minimal length addition chain is not easy; a generalized version of the problem, in which one must nd a chain that simultaneously forms each of a sequence of values, is NP-complete. For the special primes used in such cryptosystems, nding fast add...

The Diffie-Hellman problem as a cryptographic primitive plays an important role in modern cryptology. The Bit Security or Hard-Core Bits of Diffie-Hellman problem in arbitrary finite cyclic group is a long-standing open problem in cryptography. Until now, only few groups have been studied. Hyperelliptic curve cryptography is an alternative to elliptic curve cryptography. Due to the recent crypt...

This paper explores two techniques on a family of hyperelliptic curves that have been proposed to accelerate computation of scalar multiplication for hyperelliptic curve cryptosystems. In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, the τ -adic non-adjacent form (τ -NAF). It is shown that the τ -NAF has the three properties: (1) existen...

It is essential to secure the implementation of cryptosystems in embedded devices agains side-channel attacks. Namely, in order to resist differential (DPA) attacks, randomization techniques should be employed to decorrelate the data processed by the device from secret key parts resulting in the value of this data. Among the countermeasures that appeared in the literature were those that result...

In this paper, a new blind identity-based signature scheme with message recovery based on bilinear pairings on elliptic curves is presented. The work is motivated by the importance of blind signatures as a cryptographic primitive essential in protocols that guarantee anonymity of users. This is particularly of interest in DRM systems, electronic cash systems, electronic voting systems and locat...

In this paper, we propose a blind signature scheme and three practical educed schemes based on elliptic curve discrete logarithm problem. The proposed schemes impart the GOST signature structure and utilize the inherent advantage of elliptic curve cryptosystems in terms of smaller key size and lower computational overhead to its counterpart public key cryptosystems such as RSA and ElGamal. The ...

The major building block of most elliptic curve cryptosystems are computation of multi-scalar multiplication. This paper proposes a novel algorithm for simultaneous multi-scalar multiplication, that is by employing addition chains. The previously known methods utilizes double-and-add algorithm with binary representations. In order to accomplish our purpose, an efficient empirical method for fin...