Shor’s quantum algorithm for discrete logarithms applied to elliptic curve groups forms the basis of a “quantum attack” of elliptic curve cryptosystems. To implement this algorithm on a quantum computer requires the efficient implementation of the elliptic curve group operation. Such an implementation requires we be able to compute inverses in the underlying field. In [PZ03], Proos and Zalka sh...

We demonstrate a fundamental flaw in the present definition of embedding degree for curves of any genus, and we present examples of elliptic curves and genus 2 curves which highlight the error. We explain how this can yield a dramatic (unbounded) difference between the size of the actual and presumed embedding fields. This observation has serious implications for the security of pairing-based c...

This paper presents an overview of the use of elliptic curves in cryptography. The security of this cryptosystem is based on the discrete logarithm problem, which appears to be much harder compared to the discrete logarithm problem in other cryptosystems. An overview of common cryptosystems is given, such as Diffie-Hellman and RSA, and an elliptic curve cryptography scheme is discussed. -------...

This abstract presents results on polynomial systems involved in an algebraic attack on elliptic curves cryptosystems. The security of these cryptosystems is based on the difficulty to solve the elliptic curves discrete logarithm problem (ECDLP): let E be an elliptic curve defined over a finite field K. The set of its rational points forms a commutative group, E(K). Given two points P and Q of ...

Since ancient times, there has been a tug-of-war taking place between code makers and code breakers. Only within the last fifty years have the code makers emerged victorious (for now that is) with the advent of public key cryptography. This paper surveys the mathematical foundations, shortcomings, and novel variants of the “first” public key cryptosystem envisioned by Whitfield Diffie, Martin H...

Possibility of the emergence of quantum computers in the near future, pose a serious threat against the security of widely-used public key cryptosystems such as RSA or Elliptic Curve Cryptography (ECC). Algorithms involving isogeny computations on supersingular elliptic curves have been shown to be difficult to break, even to quantum computers. Thus, isogeny-based protocols represent promising ...

This work proposes a processor architecture for elliptic curves cryptosystems over fields GF (2). This is a scalable architecture in terms of area and speed that exploits the abilities of reconfigurable hardware to deliver optimized circuitry for different elliptic curves and finite fields. The main features of this architecture are the use of an optimized bitparallel squarer, a digit-serial mu...

Our purpose is to describe elliptic curves with complex multiplication which in characteristic 2 have the following useful properties for constructing Diffie-HeUman type cryptosystems: (1) they are nonsupersingular (so that one cannot use the Menezes-Okamoto-Vanstone reduction of discrete log from elliptic curves to finite fields); (2) the order of the group h a s a large prime factor (so that ...

Efficient and high-performance implementation of point multiplication is crucial for elliptic curve cryptosystems. In this paper, we present a new double point multiplication algorithm based on differential addition chains. We use our scheme to implement single point multiplication on binary elliptic curves with efficiently computable endomorphisms. Our proposed scheme has a uniform structure a...