19,860 research outputs found
The Rabin cryptosystem revisited
The Rabin public-key cryptosystem is revisited with a focus on the problem of
identifying the encrypted message unambiguously for any pair of primes. In
particular, a deterministic scheme using quartic reciprocity is described that
works for primes congruent 5 modulo 8, a case that was still open. Both
theoretical and practical solutions are presented. The Rabin signature is also
reconsidered and a deterministic padding mechanism is proposed.Comment: minor review + introduction of a deterministic scheme using quartic
reciprocity that works for primes congruent 5 modulo
Fermat quotients: Exponential sums, value set and primitive roots
For a prime and an integer with , we define Fermat
quotients by the conditions D. R. Heath-Brown has given a bound of
exponential sums with consecutive Fermat quotients that is nontrivial for
for any fixed . We use a recent idea of M.
Z. Garaev together with a form of the large sieve inequality due to S. Baier
and L. Zhao, to show that on average over one can obtain a nontrivial
estimate for much shorter sums starting with . We also
obtain lower bounds on the image size of the first consecutive Fermat
quotients and use it to prove that there is a positive integer such that is a primitive root modulo
Computing the cardinality of CM elliptic curves using torsion points
Let E be an elliptic curve having complex multiplication by a given quadratic
order of an imaginary quadratic field K. The field of definition of E is the
ring class field Omega of the order. If the prime p splits completely in Omega,
then we can reduce E modulo one the factors of p and get a curve Ep defined
over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a
fast way to find this sign. For this, we propose to use the action of the
Frobenius on torsion points of small order built with class invariants a la
Weber, in a manner reminiscent of the Schoof-Elkies-Atkin algorithm for
computing the cardinality of a given elliptic curve modulo p. We apply our
results to the Elliptic Curve Primality Proving algorithm (ECPP).Comment: Revised and shortened version, including more material using
discriminants of curves and division polynomial
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