Cryptographic protocols often make use of the inherent hardness of the classical discrete logarithm problem, which is to solve gx ≡ y (mod p) for x. The hardness of this problem has been exploited in the Diffie-Hellman key exchange, as well as in cryptosystems such as ElGamal. There is a similar discrete logarithm problem on elliptic curves: solve kB = P for k. Therefore, Diffie-Hellman and ElG...

In recent years cryptographic protocols based on the Weil and Tate pairings on elliptic curves have attracted much attention. A notable success in this area was the elegant solution by Boneh and Franklin [7] of the problem of efficient identity-based encryption. At the same time, the security standards for public key cryptosystems are expected to increase, so that in the future they will be cap...

This report introduces a new class of Galois eld used to achieve fast nite eld arithmetic which we call an Optimal Extension Field (OEF). This approach is well suited for implementation of public-key cryptosystems based on elliptic and hyperelliptic curves on RISC workstations. We de ne OEFs and describe methods for their construction. In addition, we demonstrate that use of an OEF yields the f...

Elliptic Curve cryptosystems appear to be more secure and efficient when requiring small key size to implement than other public key cryptosystems. Its security is based upon the difficulty of solving Elliptic Curve Discrete Logarithm Problem (ECDLP). This study proposes a variant of generic algorithm Pollard’s Rho for finding ECDLP using cycle detection with stack and a mixture of cycle detect...

Elliptic curves over the ring Z / n Z where R is the product of two large primes have first been proposed for public key cryptosystems in [4]. The security of this system is based on the integer factorization problem, but it is unknown whether breaking the system is equivalent to factoring. In this paper, we present a variant of this cryptosystem for which breaking the system is equivalent to f...

The origin of pairing based cryptosystems is in the MOV attack [10] on the elliptic curve discrete logarithm problem. The attack was first envisioned by Gerhard Frey. The idea was to use the bilinear properties of the Weil pairing to reduce a discrete logarithm problem in an elliptic curve over a finite field Fq to a discrete logarithm problem in Fqk . It is known [1] that most of the time for ...

Elliptic curves over the ring Z Z=nZ Z where n is the product of two large primes have rst been proposed for public key cryptosystems in 4]. The security of this system is based on the integer factorization problem, but it is unknown whether breaking the system is equivalent to factoring. In this paper, we present a variant of this cryptosystem for which breaking the system is equivalent to fac...

For elliptic curve cryptosystems does exist many algorithms, that computes the scalar multiplication k·P. Some are better for a software solution and others are better for a hardware solution. In this paper we compare algorithms without precomputation for the scalar multiplication on elliptic curves over a finite field of GF(2). At the end we show which algorithm is the best for a hardware or s...

Elliptic Curve Cryptography (ECC) can be used as a tool for encrypting data, creating digital signatures, and performing key exchanges. Regarding the encryption capability, the first schemes that appeared were either versions of already existing public-key algorithms (Massey-Omura and ElGamal) or new schemes specified ad hoc (MenezesVanstone). However, all the initial elliptic curve encryption ...

This paper presents new research results concerning cryptosystems based upon elliptic curves. The system which has been developed is similar to that of ElGamal [4] over a finite field Fq [7]. The investigations cover those elliptic curves with primary characteristics over Fq (216 q 2257, q is a prime number). The aim of the study is the determination of dependencies between calculation time req...