25,565 research outputs found
Short Polynomial Representations for Square Roots Modulo p
Abstract. Let p be an odd prime number and a a square modulo p. It is well known that the simple formula a p+1 4 mod p gives a square root of a when p ≡ 3 mod 4. Let us write p − 1 = 2 n s with s odd. A fast algorithm due to Shanks, with n steps, allows us to compute a square root of a modulo p. It will be shown that there exists a polynomial of at most 2 n−1 terms giving a square root of a. Moreover, if there exists a polynomial in a representing a square root of a modulo p, it will be proved that this polynomial would have at least 2 n−1 terms, except for a finite set P n of primes p depending on n. Résumé. Soit p un nombre premier impair et a un carré modulo p. La formule très simple a p+1 4 mod p fournit une valeur de la racine carrée de a lorsque p ≡ 3 mod 4. Plus généralement, si l'onécrit p − 1 = 2 n s avec s impair, un algorithme dûà Shanks, comprenant nétapes, permet de calculer la racine carrée de a modulo p. Nous montrerons qu'il existe un polynôme d'au plus 2 n−1 termes et dont la valeur est une racine carrée de a pour tout carré a. De plus, pour n fixé, nous démontrons que tout polynôme en a représentant la racine carrée de a modulo p a au moins 2 n−1 termes, excepté pour un ensemble fini P n de nombres premiers p ≡ 1 (mod 2 n )
Iwasawa theory and the Eisenstein ideal
In this paper, we relate three objects. The first is a particular value of a
cup product in the cohomology of the Galois group of the maximal unramified
outside p extension of a cyclotomic field containing the pth roots of unity.
The second is an Iwasawa module over a nonabelian extension of the rationals, a
subquotient of the maximal pro-p abelian unramified completely split at p
extension of a certain pro-p Kummer extension of a cyclotomic field that
contains all p-power roots of unity. The third is the quotient of an Eisenstein
ideal in an ordinary Hecke algebra of Hida by the square of the Eisenstein
ideal and the element given by the pth Hecke operator minus one. For the
relationship between the latter two objects, we employ the work of Ohta, in
which he considered a certain Galois action on an inverse limit of cohomology
groups to reestablish the Main Conjecture (for p at least 5) in the spirit of
the Mazur-Wiles proof. For the relationship between the former two objects, we
construct an analogue to the global reciprocity map for extensions with
restricted ramification. These relationships, and a computation in the Hecke
algebra, allow us to prove an earlier conjecture of McCallum and the author on
the surjectivity of a pairing formed from the cup product for p < 1000. We give
one other application, determining the structure of Selmer groups of the
modular representation considered by Ohta modulo the Eisenstein ideal.Comment: 37 page
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
Square-full primitive roots
We use character sum estimates to give a bound on the least square-full
primitive root modulo a prime. Specifically, we show that there is a
square-full primitive root mod less than , and we give some conditional bounds.Comment: 9 page
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