On Modular Inverses of Cyclotomic Polynomials and the Magnitude of their Coefficients

Let p and r be two primes and n, m be two distinct divisors of pr. Consider the n-th and m-th cyclotomic polynomials. In this paper, we present lower and upper bounds for the coefficients of the inverse of one of them modulo the other one. We mention an application to torus-based cryptography.Comment: 21 page

Normal Elliptic Bases and Torus-Based Cryptography

We consider representations of algebraic tori $T_n(F_q)$ 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 $m$ $\phi(n)$-tuples of $F_q$ 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

Computing sparse multiples of polynomials

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t non-zero terms, and if so, to find such an h. When F=Q and t is constant, we give a polynomial-time algorithm in d and the size of coefficients in h. When F is a finite field, we show that the problem is at least as hard as determining the multiplicative order of elements in an extension field of F (a problem thought to have complexity similar to that of factoring integers), and this lower bound is tight when t=2.Comment: Extended abstract appears in Proc. ISAAC 2010, pp. 266-278, LNCS 650