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.

Let p and r be two primes, and let n and m be two distinct divisors of pr. Consider Φn and Φm, the nth and mth cyclotomic polynomials. In this paper, we present lower and upper bounds for the coefficients of the inverse of Φn modulo Φm and discuss an application to torus-based cryptography.

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