Results 1  10
of
28
On some computational problems in finite abelian groups
 Mathematics of Computation
, 1997
"... Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the Oconstants and Ωconstants. We implemented the algorithms for class ..."
Abstract

Cited by 23 (7 self)
 Add to MetaCart
Abstract. We present new algorithms for computing orders of elements, discrete logarithms, and structures of finite abelian groups. We estimate the computational complexity and storage requirements, and we explicitly determine the Oconstants and Ωconstants. We implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results. Our algorithms are based on a modification of Shanks ’ babystep giantstep strategy, and have the advantage that their computational complexity and storage requirements are relative to the actual order, discrete logarithm, or size of the group, rather than relative to an upper bound on the group order. 1.
Short Representation of Quadratic Integers
 PROCEEDINGS OF CANT
, 1992
"... Let O be a real quadratic order of discriminant \Delta. For elements ff in O we develop a compact representation whose binary length is polynomially bounded in log log H(ff), log N(ff) and log \Delta where H(ff) is the height of ff and N(ff) is the norm of ff. We show that using compact representa ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
Let O be a real quadratic order of discriminant \Delta. For elements ff in O we develop a compact representation whose binary length is polynomially bounded in log log H(ff), log N(ff) and log \Delta where H(ff) is the height of ff and N(ff) is the norm of ff. We show that using compact representations we can in polynomial time compute norms, signs, products, and inverses of numbers in O and principal ideals generated by numbers in O. We also show how to compare numbers given in compact represention in polynomial time.
Some cases of the FontaineMazur conjecture
 J. Number Theory
, 1992
"... Abstract. We prove more special cases of the FontaineMazur conjecture regarding padic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], ..."
Abstract

Cited by 12 (3 self)
 Add to MetaCart
Abstract. We prove more special cases of the FontaineMazur conjecture regarding padic Galois representations unramified at p, and we present evidence for and consequences of a generalization of it. 0. Introduction. Answering a longstanding question of Furtwängler, mentioned as early as 1926 [10], Golod and Shafarevich showed in 1964 [8] that there exists a number field with an infinite, everywhere unramified prop extension. In fact it is easy to obtain many examples [8], [13], [23]. Very little is known, however, regarding the structure
Order computations in generic groups
 PHD THESIS MIT, SUBMITTED JUNE 2007. RESOURCES
, 2007
"... ..."
A method to solve cyclotomic norm equations
 Algorithmic number theory: 6th international symposium  ANTSVI
, 2004
"... Abstract. We present a technique to recover f ∈ Q(ζp) where ζp is a primitive pth root of unity for a prime p, given its norm g = f ∗ ¯ f in the totally real field Q(ζp + ζ −1 p). The classical method of solving this problem involves finding generators of principal ideals by enumerating the whole c ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. We present a technique to recover f ∈ Q(ζp) where ζp is a primitive pth root of unity for a prime p, given its norm g = f ∗ ¯ f in the totally real field Q(ζp + ζ −1 p). The classical method of solving this problem involves finding generators of principal ideals by enumerating the whole class group associated with Q(ζp), but this approach quickly becomes infeasible as p increases. The apparent hardness of this problem has led several authors to suggest the problem as one suitable for cryptography. We describe a technique which avoids enumerating the class group, and instead recovers f by factoring Nf, the absolute norm of f, (for example with a subexponential sieve algorithm), and then running the GentrySzydlo polynomial time algorithm for a number of candidates. The algorithm has been tested with an implementation in PARI. 1
HEURISTICS FOR CLASS NUMBERS AND LAMBDA INVARIANTS
"... Abstract. Let K = Q ( √ −d) be an imaginary quadratic field and let Q ( √ 3d) be the associated real quadratic field. Starting from the CohenLenstra heuristics and Scholz’s theorem, we make predictions for the behaviors of the 3parts of the class groups of these two fields as d varies. We deduce ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. Let K = Q ( √ −d) be an imaginary quadratic field and let Q ( √ 3d) be the associated real quadratic field. Starting from the CohenLenstra heuristics and Scholz’s theorem, we make predictions for the behaviors of the 3parts of the class groups of these two fields as d varies. We deduce heuristic predictions for the behavior of the Iwasawa λinvariant for the cyclotomic Z3extension of K and test them computationally. The CohenLenstra heuristics [1] give predictions for frequencies of class numbers and class groups of number fields. In the following, we investigate a related situation and a more specific question: I. Are there heuristics for the Iwasawa lambda invariants, similar to those of Cohen and Lenstra for class groups of number fields? The λ2invariants of imaginary quadratic fields are given by a simple formula of Ferrero [4] and Kida [5] and are correspondingly not suitable for a heuristic analysis. We therefore consider the first nontrivial case, namely the λinvariant for the cyclotomic Z3extension of an imaginary quadratic field K as K varies. When 3 does not split in K, the frequency
On the computation of the class numbers of real abelian fields
 Turku Centre for Computer Science
, 2007
"... Abstract. In this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes < 10000 not dividing the degree that divide the class numbers of fields of conductor ≤ 2000. Cohen–Lenstra heuristics allow us to conjecture that no larg ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Abstract. In this paper we give a procedure to search for prime divisors of class numbers of real abelian fields and present a table of odd primes < 10000 not dividing the degree that divide the class numbers of fields of conductor ≤ 2000. Cohen–Lenstra heuristics allow us to conjecture that no larger prime divisors should exist. Previous computations have been largely limited to prime power conductors. 1.
Arithmetic Properties of Class Numbers of Imaginary Quadratic Fields
, 2006
"... Under the assumption of the wellknown heuristics of Cohen and Lenstra (and the new extensions we propose) we give proofs of several new properties of class numbers of imaginary quadratic number fields, including theorems on smoothness and normality of their divisors. Some applications in cryptograp ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Under the assumption of the wellknown heuristics of Cohen and Lenstra (and the new extensions we propose) we give proofs of several new properties of class numbers of imaginary quadratic number fields, including theorems on smoothness and normality of their divisors. Some applications in cryptography are also discussed. 1