Computing discrete logarithms takes time. It takes time to develop new algorithms, choose the best algorithms, implement these algorithms correctly and efficiently, keep the system running for several months, and, finally, publish the results. In this paper, we present a highly performant architecture that can be used to compute discrete logarithms of Weierstrass curves defined over binary fiel...

The MOV and FR algorithms, which are representative attacks on elliptic curve cryptosystems, reduce the elliptic curve discrete logarithm problem (ECDLP) to the discrete logarithm problem in a finite field. This paper studies these algorithms and introduces the following three results. First, we show an explicit condition under which the MOV algorithm can be applied to non-supersingular ellipti...

Zero-knowledge proof protocol is a basic cryptographic technique. And zero-knowledge proof of double discrete logarithm has some particular properties, so it has been widely applied in many security systems. But the efficient problem of zero-knowledge proof of double discrete logarithm has not been solved to this day, since there are some special difficulties in computing this kind of knowledge...

In 2002, Hwang et al. propose an asymmetric cryptosystem in which a large message is broken into smaller plaintexts which are then encrypted efficiently. They claim that based on the assumption of the intractability of the discrete logarithm problem, their cryptosystem is secure in the sense that even knowing some pairs of plaintext-ciphertext does not allow an intruder to acquire other plainte...

The Elliptic Curve Cryptosystem (ECC) provides the highest strength-per-bit of any cryptosystem known today. This paper provides an overview of the three hard mathematical problems which provide the basis for the security of publickey cryptosystems used today: the integer factorization problem (IFP), the discrete logarithm problem (DLP), and the elliptic curve discrete logarithm problem (ECDLP)...