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Factoring integers with the number field sieve
In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the “number field sieve”, which was proposed by John Pollard. The present paper is devoted to the description
Arithmetic on elliptic curves with complex multiplication. II
0. In this paper we will continue to study the arithmetic of elliptic curves with complex multiplication by Q ( 1 / ~ ) , which we began in [5]. Chapter I reviews the basic facts on Q-curves, and
On the essential dimension of a finite group
Let f(x) = Σaixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for
Juggling Drops and Descents
(1994). Juggling Drops and Descents. The American Mathematical Monthly: Vol. 101, No. 6, pp. 507-519.
On the Evaluation of Euler Sums
Various series expansions of ζ(r, s) for real numbers r and s are established, which generally involve infinitely many zeta values.
Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography
1. Solving Pell's equation Hendrik Lenstra 2. Basic algorithms in number theory Joe Buhler and Stan Wagon 3. Elliptic curves Bjorn Poonen 4. The arithmetic of number rings Peter Stevenhagen 5. Fast
Lattice basis reduction, Jacobi sums and hyperelliptic cryptosystems
  • J. Buhler, N. Koblitz
  • Mathematics, Computer Science
    Bulletin of the Australian Mathematical Society
  • 1 August 1998
Using the LLL-algorithm for finding short vectors in lattices, it is shown how to compute a Jacobi sum for the prime field Fp in Q(e2πi/n) in time O(log3p), useful in the construction of hyperelliptic cryptosystems.
On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3
The elliptic curve y2 = 4x3 28x + 25 has rank 3 over Q. Assuming the WeilTaniyama conjecture for this curve, we show that its L-series L(s) has a triple zero at s = 1 and compute lim, _I L(s)/(s 1)3
Heuristics for class numbers of prime-power real cyclotomic fields
Abstract. Let h(`) denote the class number of the maximal totally real subfield Q(cos(2π/`n)) of the field of `n-th roots of unity. The goal of this paper is to show that (speculative extensions of)