Takakazu Satoh

On Certain Vector Valued Siegel Modular Forms of Degree Two

In this paper we explicitly construct vector valued Siegel modular forms of degree two and the automorphic factor detk| for even k where St denotes the standard representation of GL(2,C). As an application, we prove some congruences between eigenvalues of Hecke operators. For a positive integer n, let F, be the full Siegel modular group of degree n and H, the Siegel upper half plane of degree n. For M ~ F, and Z ~ H,, we put

Some Remarks on Triple L-Functions.

We extend the Garretts result I-4] concerning triple L-function in two aspects. One concerns triple L-functions of different weights and the other concerns special values of triple L-functions of same weights at inner points of their critical strips. Let Mk(Fn) and Sk(Fn) be spaces of Siegel modular forms and cuspforms of weight k and of degree n, respectively. For Siegel modular forms f l , .--,fro (of various degree) and a field K, we denote by K(ft,..., fro) the field generated by all the Fourier coefficients of f l , .-.,fro over K. We write Fourier expansion of f~ Sk(F1)

Generating Genus Two Hyperelliptic Curves over Large Characteristic Finite Fields

In hyperelliptic curve cryptography, finding a suitable hyperelliptic curve is an important fundamental problem. One of necessary conditions is that the order of its Jacobian is a product of a large prime number and a small number. In the paper, we give a probabilistic polynomial time algorithm to test whether the Jacobian of the given hyperelliptic curve of the form Y 2 = X 5 + u X 3 + v X satisfies the condition and, if so, to give the largest prime factor. Our algorithm enables us to generate random curves of the form until the order of its Jacobian is almost prime in the above sense. A key idea is to obtain candidates of its zeta function over the base field from its zeta function over the extension field where the Jacobian splits.

On pairing inversion problems

In many aspects, cryptanalyses of pairing based cryptography consider protocol level security and take difficulties of primitives for granted. In this survey, we consider pairing inversion. At the time this manuscript was written(April 2007), to the best of the authors knowledge, there are neither known feasible algorithms for pairing inversions nor published proofs that the problem is unfeasible.

On degrees of polynomial interpolations related to elliptic curve cryptography

We study two topics on degrees of polynomials which interpolate cryptographic functions. The one is concerned with elliptic curve discrete logarithm (ECDL) on curves with an endomorphism of degree 2 or 3. For such curves, we obtain a better lower bound of degrees for polynomial interpolation of ECDL. The other deals with degrees of polynomial interpolations of embeddings of a subgroup of the multiplicative group of a finite field to an elliptic curve.

Closed formulae for the Weil pairing inversion

Using the Miller algorithm, we can efficiently compute the Weil pairing for two given points on an elliptic curve. On the other hand, security of pairing based cryptographic protocols depends on the converse problem: find a point on an elliptic curve whose Weil pairing with a given (fixed) point is equal to a given root of unity, which we call the Weil pairing inversion problem. In this article, we give closed formulae which give a solution to the problem. For supersingular elliptic curves over fields of characteristic two or three, these formulae take more simpler forms than those for other elliptic curves.

On the optimality of lattices for the coppersmith technique

We investigate the Coppersmith technique [7] for finding solutions of a univariate modular equation within a range given by range parameter U. This paper provides a way to analyze a general type of limitation of the lattice construction. Our analysis bounds the possible range of U from above that is asymptotically equal to the bound given by the original result of Coppersmith. To show our result, we establish a framework for the technique by following the reformulation of Howgrave-Graham [14], and derive a condition for the technique to work. We then provide a way to analyze a bound of U for achieving the condition. Technically, we show that (i) the original result of Coppersmith achieves an optimal bound for U when constructing a lattice in a standard way. We then show evidence supporting that (ii) a non-standard lattice construction is generally difficult. We also report on computer experiments demonstrating the tightness of our analysis. Some of the detailed arguments are omitted due to the space limit; see the full-version [1].

More Discriminants with the Brezing-Weng Method

The Brezing-Weng method is a general framework to generate families of pairing-friendly elliptic curves. Here, we introduce an improvement which can be used to generate more curves with larger discriminants. Apart from the number of curves this yields, it provides an easy way to avoid endomorphism rings with small class number.

Euclid Prime Sequences over Unique Factorization Domains

The proof by Euclid that there exist infinitely many prime numbers is well known. The proof involves generating prime numbers that do not belong to a given finite set of primes, and one may ask whether all prime numbers can be obtained by this method. Daniel Shanks gave a heuristic argument that suggests that the answer is affirmative. Despite recent advances in computational number theory, numerical examples do not seem to make this conjecture convincing. We reformulate the problem in polynomial rings over finite fields and prove that in some explicitly characterized cases, Shankss argument does not hold. On the other hand, we have performed numerical computations that suggest that except for the above cases, Shankss conjecture is true.

Vector Valued Modular Forms of Degree Two and their Application to Triple L-functions

Abstract We report some properties of vector valued Siegel modular forms of degree two and triple L-functions of different weight. Their relation is discussed in the section 3. This is motivated by an observation in the section 2. In both sections, certain differential operators are relevant. They are defined in the section 1.