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On the lines of the binary gcd algorithm for rational integers, algorithms for computing the gcd are presented for the ring of integers in $\mathbb{Q}(\sqrt{d})$ where d ∈ { − 2, − 7, − 11, − 19}. Thus a binary gcd like algorithm is presented for a unique factorization domain which is not Euclidean (case d=-19). Together with the earlier known binary gcd like algorithms for the ring of integers...
Given an algebraic number field K, such that [K : ℚ] is constant, we show that the problem of computing the units group is in the complexity class SPP. As a consequence, we show that principal ideal testing for an ideal in is in SPP. Furthermore, assuming the GRH, the class number of K, and a presentation for the class group of K can also be computed in SPP...
We provide explicit formulae for realising the group law in Jacobians of superelliptic curves of genus 3 and C3,4 curves. It is shown that two distinct elements in the Jacobian of a C3,4 curve can be added with 150 multiplications and 2 inversions in the field of definition of the curve, while an element can be doubled with 174 multiplications and 2 inversions. In superelliptic...
The recent ideas of Agrawal, Kayal, and Saxena have produced a milestone in the area of deterministic primality testing. Unfortunately, their method, as well as their successors are mainly of theoretical interest, as they are much too slow for practical applications. Via a totally different approach, Lukes et al. developed a test which is conjectured to prove the primality of N in time only...
We discuss the situation where a curve , defined over a number field K, has a known K-rational divisor class of degree 1, and consider whether this class contains an actual K-rational divisor. When has points everywhere locally, the local to global principle of the Brauer group gives the existence of such a divisor. In this situation, we give an alternative, more down...
In this paper, we study p-divisibility of discriminants of Hecke algebras associated to spaces of cusp forms of prime level. By considering cusp forms of weight bigger than 2, we are are led to make a precise conjecture about indexes of Hecke algebras in their normalisation which implies (if true) the surprising conjecture that there are no mod p congruences between non-conjugate newforms in S ...
Using powerful tools on genus 2 curves like the Kummer variety, we generalize the Montgomery method for scalar multiplication to the jacobian of these curves. Previously this method was only known for elliptic curves. We obtain an algorithm that is competitive compared to the usual methods of scalar multiplication and that has additional properties such as resistance to timings attacks. This algorithm...
We present algorithms for computing the squared Weil and Tate pairings on elliptic curves and the squared Tate pairing on hyperelliptic curves. The squared pairings introduced in this paper have the advantage that our algorithms for evaluating them are deterministic and do not depend on a random choice of points. Our algorithm to evaluate the squared Weil pairing is about 20% more efficient than the...
We use rational parametrizations of certain cubic surfaces and an explicit formula for descent via 3-isogeny to construct the first examples of elliptic curves Ek: x3 + y3 = k of ranks 8, 9, 10, and 11 over ℚ. As a corollary we produce examples of elliptic curves over ℚ with a rational 3-torsion point and rank as high as 11. We also discuss...
The elliptic curve primality proving algorithm is one of the fastest practical algorithms for proving the primality of large numbers. Its fastest version, fastECPP, runs in heuristic time $\widetilde{O}(({\rm log}N)^{4})$ . The aim of this article is to describe new ideas used when dealing with very large numbers. We illustrate these with the primality proofs of some numbers with more than 10,000...
We present an algorithm based on the birthday paradox, which is a low-memory parallel counterpart to the algorithm of Matsuo, Chao and Tsujii. This algorithm computes the group order of the Jacobian of a genus 2 curve over a finite field for which the characteristic polynomial of the Frobenius endomorphism is known modulo some integer. The main tool is a 2-dimensional pseudo-random walk that allows...
In this paper we investigate the efficiency of the function field sieve to compute discrete logarithms in the finite fields . Motivated by attacks on identity based encryption systems using supersingular elliptic curves, we pay special attention to the case where n is composite. This allows us to represent the function field over different base fields. Practical experiments appear...
We give a comparison of the performance of the recently proposed torus-based public key cryptosystem CEILIDH, and XTR. Underpinning both systems is the mathematics of the two dimensional algebraic torus $T_{6}(\mathbb{F}_{p})$ . However, while they both attain the same discrete logarithm security and each achieve a compression factor of three for all data transmissions, the arithmetic performed...
We introduce a new model for elliptic curves over rings of odd characteristic, and study its properties and its utility in numerical computations. They turn to be particularly interesting for elliptic curves with complex multiplication, for which they provide very simple stable equations. The invariants associated to these models allow an easy construction of ring class fields of certain imaginary...
We develop an algorithm for computing isomorphisms and automorphisms of algebraic function fields of transcendence degree one in characteristic zero and positive characteristic.
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 in the totally real field $\mathbb{Q}(\zeta_{p}+\zeta_{p}^{-1})$ . The classical method of solving this problem involves finding generators of principal...
It is well-known that the minus class number h of an imaginary cyclic quartic field of prime conductor p can grow arbitrarily large, but until now no one has been able to exhibit an example for which h > p. In an attempt to find such an example, we have tabulated h for all primes p ≡ 5(mod 8) with p < 1010 and for primes p < 1014 satisfying...
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