Normal elliptic bases and torus-based cryptography

Abstract

We consider representations of algebraic tori Tn(Fq) over finite fields. We make use of normal elliptic bases to show that, for infinitely many squarefree integers n and infinitely many values of q, we can encode m torus elements, to a small fixed overhead and to m ϕ(n)-tuples of Fq elements, in quasi-linear time in log q.This improves upon previously known algorithms, which all have a quasi-quadratic complexity. As a result, the cost of the encoding phase is now negligible in Diffie-Hellman cryptographic schemes.

On modular inverses of cyclotomic polynomials and the magnitude of their coefficients

“…In this section we will briefly describe how Theorem 1 has been used in [12]. We refer to the latter paper for more details and further references.…”

“…In [12] we present a practical implementation of this map, whose efficiency relies on the use of a certain class of normal bases (see [10]) in the representation of field extensions.…”

“…Beyond the simple arithmetic context of this computation, we actually found a direct application in torus-based cryptography. In this section we will briefly present the use of Theorem 1 that has been made in [12]. We refer to the latter for more details and more references.…”

“…Such calculations typically occur in torus-based cryptographic schemes, as developed by Silverberg and Rubin [19,20]. The bounds presented in Theorem 1 lead to improvements in the running times of algorithms in this field (see [11,12]). Such schemes are discrete log-based cryptosystems and make use of a subgroup of F × q n in which the communication cost is reduced; that is to say, elements can be represented by fewer than the usual n coordinates in F q .…”

“…We make use of a certain class of normal bases [10], which allows efficient arithmetic in F q n . See [12] for more details.…”