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18
Multidigit Multiplication For Mathematicians
, 2001
"... This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStr ..."
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Cited by 31 (9 self)
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This paper surveys techniques for multiplying elements of various commutative rings. It covers Karatsuba multiplication, dual Karatsuba multiplication, Toom multiplication, dual Toom multiplication, the FFT trick, the twisted FFT trick, the splitradix FFT trick, Good's trick, the SchönhageStrassen trick, Schönhage's trick, Nussbaumer's trick, the cyclic SchönhageStrassen trick, and the CantorKaltofen theorem. It emphasizes the underlying ring homomorphisms.
Do all elliptic curves of the same order have the same difficulty of discrete log
 Advances in Cryptology — ASIACRYPT 2005, Lecture Notes in Computer Science
"... Abstract. The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (dlog) has the same difficulty for all curves over a given finite field with ..."
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Cited by 14 (4 self)
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Abstract. The aim of this paper is to justify the common cryptographic practice of selecting elliptic curves using their order as the primary criterion. We can formalize this issue by asking whether the discrete log problem (dlog) has the same difficulty for all curves over a given finite field with the same order. We prove that this is essentially true by showing polynomial time random reducibility of dlog among such curves, assuming the Generalized Riemann Hypothesis (GRH). We do so by constructing certain expander graphs, similar to Ramanujan graphs, with elliptic curves as nodes and low degree isogenies as edges. The result is obtained from the rapid mixing of random walks on this graph. Our proof works only for curves with (nearly) the same endomorphism rings. Without this technical restriction such a dlog equivalence might be false; however, in practice the restriction may be moot, because all known polynomial time techniques for constructing equal order curves produce only curves with nearly equal endomorphism rings.
conjectures and Hilbert’s twelfth problem. Experiment
 Math. 9
, 1996
"... We give a constructive proof of a theorem given in [Tate 84] which states that (under Stark’s Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real Abelian extension of K. As a direct application of this proof, we show how one can compute explicitly ..."
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Cited by 11 (8 self)
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We give a constructive proof of a theorem given in [Tate 84] which states that (under Stark’s Conjecture) the field generated over a totally real field K by the Stark units contains the maximal real Abelian extension of K. As a direct application of this proof, we show how one can compute explicitly real Abelian extensions of K. We give two examples. In a series of important papers [Stark 71, Stark 75, Stark 76, Stark 80] H. M. Stark developed a body of conjectures relating the values of Artin Lfunctions at s = 1 (and hence, by the functional equation, their leading terms at s = 0) with certain algebraic quantities attached to extensions of number fields. For example, in the case of Abelian Lfunctions with a firstorder zero at s = 0, the conjectural relation is between the first derivative of the Lfunctions and the logarithmic embedding of certain units in ray class fields known as Stark units, which are predicted to exist. The use of these conjectures to provide explicit generators of ray class fields,
Interpolation of ShiftedLacunary Polynomials (Extended Abstract)
"... Given a “black box” function to evaluate an unknown rational polynomial f ∈Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t∈Z>0, the shift α∈Q, the exponents 0≤e1< ..."
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Cited by 9 (1 self)
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Given a “black box” function to evaluate an unknown rational polynomial f ∈Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t∈Z>0, the shift α∈Q, the exponents 0≤e1< e2<···<et, and the coefficients c1,...,ct∈Q\{0} such that f (x)=c1(x−α) e1 + c2(x−α) e2 +···+ct(x−α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size and in particular is logarithmic in deg f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information. We give both an unconditional deterministic algorithm which is polynomialtime but has a rather high complexity, and a more practical probabilistic algorithm which relies on some unknown constants.
Computing automorphisms of abelian number fields
 Math. Comput
, 1999
"... Abstract. Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x)ofαover L. In practice, the existing algorithms for factori ..."
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Cited by 7 (5 self)
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Abstract. Let L = Q(α) be an abelian number field of degree n. Most algorithms for computing the lattice of subfields of L require the computation of all the conjugates of α. This is usually achieved by factoring the minimal polynomial mα(x)ofαover L. In practice, the existing algorithms for factoring polynomials over algebraic number fields can handle only problems of moderate size. In this paper we describe a fast probabilistic algorithm for computing the conjugates of α, which is based on padic techniques. Given mα(x) anda rational prime p which does not divide the discriminant disc(mα(x)) of mα(x), the algorithm computes the Frobenius automorphism of p in time polynomial in the size of p and in the size of mα(x). By repeatedly applying the algorithm to randomly chosen primes it is possible to compute all the conjugates of α. 1.
Computing the Hilbert class field of real quadratic fields
 Math. Comp
"... Abstract. Using the units appearing in Stark’s conjectures on the values of Lfunctions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, sothatk = Q ( √ dk), and let ω ..."
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Cited by 4 (2 self)
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Abstract. Using the units appearing in Stark’s conjectures on the values of Lfunctions at s = 0, we give a complete algorithm for computing an explicit generator of the Hilbert class field of a real quadratic field. Let k be a real quadratic field of discriminant dk, sothatk = Q ( √ dk), and let ω denote an algebraic integer such that the ring of integers of k is Ok: = Z + ωZ. An important invariant of k is its class group Clk, which is, by class field theory, associated to an Abelian extension of k, the socalled Hilbert class field, denoted by Hk. This field is characterized as the maximal Abelian extension of k which is unramified at all (finite and infinite) places. Its Galois group is isomorphic to the class group Clk; hence the degree [Hk: k] istheclassnumberhk. There now exist very satisfactory algorithms to compute the discriminant, the ring of integers and the class group of a number field, and especially of a quadratic field (see [3] and [16]). For the computation of the Hilbert class field, however, there exists an efficient version only for complex quadratic fields, using complex multiplication (see [18]), and a general method for all number fields, using Kummer
Results and estimates on pseudopowers
 Math. Comp
, 1996
"... Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a power ..."
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Cited by 2 (0 self)
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Abstract. Let n be a positive integer. We say n looks like a power of 2moduloaprime pif there exists an integer ep ≥ 0 such that n ≡ 2 ep (mod p). First, we provide a simple proof of the fact that a positive integer which looks like a power of 2 modulo all but finitely many primes is in fact a powerof2. Next, we define an xpseudopower of the base 2tobeapositiveintegern that is not a power of 2, but looks like a power of 2 modulo all primes p ≤ x. Let P2(x) denote the least such n. We give an unconditional upper bound on P2(x), a conditional result (on ERH) that gives a lower bound, and a heuristic argument suggesting that P2(x)isaboutexp(c2x/log x) for a certain constant c2. We compare our heuristic model with numerical data obtained by a sieve. Some results for bases other than 2 are also given. 1.
Deciding the nilpotency of the galois group by computing elements in the centre
 Mathematics of Computation
"... Abstract. We present a new algorithm for computing the centre of the Galois group of a given polynomial f ∈ Q[x] along with its action on the set of roots of f, without previously computing the group. We show that every element in the centre is representable by a family of polynomials in Q[x]. For c ..."
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Cited by 1 (0 self)
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Abstract. We present a new algorithm for computing the centre of the Galois group of a given polynomial f ∈ Q[x] along with its action on the set of roots of f, without previously computing the group. We show that every element in the centre is representable by a family of polynomials in Q[x]. For computing such polynomials, we use quadratic Newtonlifting and truncated expressions of the roots of f over a padic number field. As an application we give a method for deciding the nilpotency of the Galois group. If f is irreducible with nilpotent Galois group, an algorithm for computing it is proposed. 1.
Numerical Verification of the BrumerStark Conjecture
"... Introduction The construction of group ring elements that annihilate the ideal class groups of totally complex abelian extensions of Q is classical and goes back to work of Kummer and Stickelberger. A generalization to totally complex abelian extensions of totally real number fields was formulated ..."
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Introduction The construction of group ring elements that annihilate the ideal class groups of totally complex abelian extensions of Q is classical and goes back to work of Kummer and Stickelberger. A generalization to totally complex abelian extensions of totally real number fields was formulated by Brumer. Brumer's formulation fits into a more general framework known as the BrumerStark conjecture. We will verify this conjecture for a large number of examples belonging to an extended class of situations where the general status of the conjecture is still unknown. We assume throughout that k is a totally real basefield and K is a totally complex extension field, abelian over k. Let wK denote the number of roots of unity in K, m = [k : Q ], and G = Gal(K=k). We also let S =<F11.23
THE SMALLEST PRIME THAT SPLITS COMPLETELY IN AN ABELIAN NUMBER FIELD
"... Abstract. Let K/Q be an abelian extension and let D be the absolute value of the discriminant of K. We show that for each ε> 0, the smallest rational prime that splits completely in K is O(D 1 4 +ε). Here the implied constant depends only on ε and the degree of K. This generalizes a theorem of El ..."
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Abstract. Let K/Q be an abelian extension and let D be the absolute value of the discriminant of K. We show that for each ε> 0, the smallest rational prime that splits completely in K is O(D 1 4 +ε). Here the implied constant depends only on ε and the degree of K. This generalizes a theorem of Elliott, who treated the case when K/Q has prime conductor. 1.