Elliptic Curve Cryptosystems

Abstract

Abstract. We discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field. These elliptic curve cryptosystems may be more secure, because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem, especially over GF(2"). We discuss the question of primitive points on an elliptic curve modulo p, and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point.1. Introduction. The earliest public key cryptosystems using number theory were based on the structure either of the multiplicative group (Z/NZ)* or the multiplicative group of a finite field G¥(q), q = p". The subsequent construction of analogous systems based on other finite Abelian groups, together with H. W. Lenstra's success in using elliptic curves for integer factorization, make it natural to study the possibility of public key cryptography based on the structure of the group of points of an elliptic curve over a large finite field. We first briefly recall the facts we need about such elliptic curves (for more details, see [4] or [5]). We then describe elliptic curve analogs of the Massey-Omura and ElGamal systems. We give some concrete examples, discuss the question of primitive points, and conclude with a theorem concerning the probability that the order of a cyclic subgroup is nonsmooth.I would like to thank A. Odlyzko for valuable discussions and correspondence, and for sending me a preprint by V. S. Miller, who independently arrived at some similar ideas about elliptic curves and cryptography.

An Algebraic Framework for Cipher Embeddings

“…There are many cryptosystems, such as ElGamal [5], the Digital Signature Standard [15] or Elliptic Curves [10,7], whose security is believed to fundamentally depend on the inability of an attacker to calculate discrete logarithms in certain finite cyclic groups. For such a group G of order p, this is equivalent to the inability of an attacker to find a presentation based on the isomorphism G → Z + p , where Z + p is the additive cyclic group of integers modulo p.…”

Efficient GF(p m ) Arithmetic Architectures for Cryptographic Applications

“…In particular, two public-key cryptosystems based on finite fields stand out: elliptic curve (EC) cryptosystems, introduced by Miller and Koblitz [24,19], and hyperelliptic cryptosystems, a generalization of elliptic curves introduced by Koblitz in [20]. Both, prime fields and extension fields, have been proposed for use in such cryptographic systems.…”

How Secure Are Elliptic Curves over Composite Extension Fields?

“…Ever since its invention, in 1986 by Koblitz [9] and Miller [12], elliptic curve cryptography (ECC) has attracted considerable interest since it enables improved security, in the sense of greater perceived strength per bit of key, compared to conventional systems such as RSA, with the added benefit of smaller key sizes, less bandwidth and less computing power, see [4] for a complete treatment of ECC. Various standards bodies, both government sponsored and industry led (for example NIST [2] and SECG [3]), have standardised on elliptic curves defined over fields of the form F 2 p and F p , where p denotes a prime.…”