Abstract. Let p be a prime and let a and b be elements of the finite field Fp of p elements. The inversive congruential generator (ICG) is a sequence (un) of pseudorandom numbers defined by the relation u n+1 ≡ au −1 n +b mod p. We show that if sufficiently many of the most significant bits of several consecutive values un of the ICG are given, one can recover the initial value u 0 (even in the case where the coefficients a and b are not known). We also obtain similar results for the quadratic congruential generator (QCG),. This suggests that for cryptographic applications ICG and QCG should be used with great care. Our results are somewhat similar to those known for the linear congruential generator (LCG), x n+1 ≡ axn + b mod p, but they apply only to much longer bit strings. We also estimate limits of some heuristic approaches, which still remain much weaker than those known for LCG.

No abstract

No abstract

We study genus 3 hyperelliptic curves which have an extra involution. The locus L 3 of these curves is a 3-dimensional subvariety in the genus 3 hyperelliptic moduli H 3 . We find a birational parametrization of this locus by affine 3-space. For every moduli point p ∈ H 3 such that |Aut(p)| > 2, the field of moduli is a field of definition. We provide a rational model of the curve over its field of moduli for all moduli points p ∈ H 3 such that |Aut(p)| > 4. This is the first time that such a rational model of these curves appears in the literature.2000 Mathematics Subject Classification. Primary 54C40, 14E20; Secondary 46E25, 20C20.

Let p be a prime and let a and b be integers modulo p. The inversive congruential generator (ICG) is a sequence (un) of pseudorandom numbers defined by the relation un+1 ≡ au −1 n + b mod p. We show that if b and sufficiently many of the most significant bits of three consecutive values un of the ICG are given, one can recover in polynomial time the initial value u0 (even in the case where the coefficient a is unknown) provided that the initial value u0 does not lie in a certain small subset of exceptional values.