Additive operation in Edwards curves is more efficient than that in elliptic curves of Weierstrass type. Therefore, Edwards curves promote the efficiency of elliptic curve cryptography. In the application to pairing-based cryptography, we need to construct an Edwards curves with prescribed order. In this talk, we explain the algorithm of the construction and Edwards curves suitable for cryptographic application.
Recently, Stange defined elliptic nets and gave an algorithm to compute the Tate pairing on an elliptic curve via elliptic nets. Elliptic nets are maps from a free Abelian group of finite rank to a ring that satisfy a certain recurrence relation. For example, elliptic divisibility sequences are a special case of elliptic nets. In this talk, we define hyperelliptic nets as a generalization of elliptic nets to hyperelliptic curves. We also describe an algorithm to compute the Tate-Lichtenbaum pairing on a curve of genus 2 by using hyperelliptic nets.
Recent simulations often need many pseudorandom number generators with distinct parameter sets, and hence we need an effective fast assessment of the generator with a given parameter set. Linear generators over the two-element field are good candidates, because of the powerful assessment via their dimensions of equidistribution. Some efficient algorithms to compute these dimensions use reduced bases of lattices associated with the generator. In this talk, we use a fast lattice reduction algorithm by Mulders and Storjohann, and show that the order of computational complexity is lessened.
In this paper, we introduce our formalization of the definitions and theorems related to an elliptic curve over a finite prime field F_p (p>3). Mizar is an advanced project of the Mizar Society which formalizes mathematics. The Mizar project, which was developed to describe mathematics formally, describes mathematical proofs in the Mizar language.
Determination problem of elliptic curves having everywhere good reduction (especially over some real and imaginary quadratic fields) is very interesting and important problem from the viewpoint of modularity conjecture. In this talk, we report recent progress including our latest result. We also explain other cases with some ideas to solve. This work is partly joint work with Yu Shimasaki (Kyushu University).
Under some assumpsions, we describe the structure of unramified Galois extensions of cyclic extensions of number fields whose Galois groups are copies of finite simple groups. We also describe application of this to $n$ th layer of the $Z_l$-extension of rational number field for small prime $l$ and $n$, and related problems.
This talk is a continuation of former ones. Let a, b be in N. We put V = { ax^2 + by^2 | x, y run through N }. H denotes the class-group of the quadratic field generated by (-ab)^(1/2). It turns out a close relation between V and H. This fact reveals several deep properties of V and H. (1) Let H be a cyclic group of odd order h. Then there is a simple algorithm to obtain h. (2) Let H be a direct product of two cyclic groups (21) and (3). Then no imaginary quadratic field allows H as its class-group.
The group G=PSL(2,29) contains two non-conjugate subgroups isomorphic to the alternating group A_5, and the pair gives inequivalent permutation representation whose ZG-module are isomorphic, as pointed out by Scott (1992). The two non-conjugate A_5's are conjugate in PGL(2,29), and one obtains a 5-regular 2-arc-transitive graph on PGL(2,29)/A_5. In this talk, we report on our computational result on the question as to which q one obtains a 5-regular 2-arc-transitive graph on PGL(2,q)/A_5 or PSL(2,q)/A_5. Also, as the permutation representation on PGL(2,29)/A_5 is multiplicity-free, we give its character table. Our computations have been done by MAGMA and GAP.
Self-dual codes are an important class of codes for both theoretical and practical reasons. In this talk, I give an introduction on self-dual codes. An algebraic approach as well as a computational approach on self-dual codes are presented.
We consider a parafermion vertex operator algebra, which is the commutant of a Heisenberg algebra in an affine vertex operator algebra of type $A_1^{(1)}$ with positive integral level. We compute the $C_2$ algebra by using a singular vector to show the $C_2$-cofiniteness of the parafermion vertex operator algebra.
Mutation is an involutive operation for quivers. If G is a tree, then the mutation class of G is finite if and only if G is a Dynkin quiver or an extended Dynkin quiver. We determined the groups generated by mutations for all n-vertex trees whose mutation classes are finite(n <= 8). Moreover, we found a mutation invariant for 3-vertex quivers and an exceptional series over Fp.
We present a simple method for determining Scott modules of finite groups using just the Brauer tree of the principal block and values of ordinary characters at non-identity p-elements. In the past it was usually necessary to find the Scott modules using direct calculation, for example when the finite group has very large order, and often a computer with a package such as GAP is needed. We classify the structures of the Scott modules and also the ordinary characters they afford by the length of the associated path in the Brauer tree.
We will study about a concrete construction of the 1333-dimensional representation of sporadic simple group $J_4$. We try to construct $J_4$ by the amalgamation of Ivanov as the 1333- dimensional modular representation over GF(3).
Let $K$ be the finite field of size $q^{2}$, and $g(F)$ (resp. $N(F)$) the genus (resp. the number of rational places) of a function field $F/K$. A recursive tower is a sequence $T= (F_{0}, F_{1}, F_{2}, \ldots)$ of function fields $F_{i}/K$ satisfying some conditions. A tower $T$ is said to be optimal if $\lim_{i \to \infty} N(F_{i})/g(F_{i})= q-1$. In 1997, Noam Elkies conjectured that every optimal recursive tower is modular. This conjecture is yet open. In this talk, we will introduce a numerical evidence of this conjecture.
From the view point of continued fraction expansions, it seems that discriminants of quadratic orders over the rational field, whose fundamental unit is relatively large, can be classified in some sense. This talk reports on a certain attempt on such classification.
A cubature formula is a numerical integration, stated as a weighted sum of function values at specified points in the domain. A typical example is the Newton-Cotes formula. In this talk we discuss an algebraic construction of cubature formulae of degree 9 and 11 on the sphere. We also show that the resulting formulae include several explicit representations of $(x_1^2 + . . . + x_d^2)^t$ in terms of linear forms of degree $2t$ with rational coefficients. Such representations stem from Waring's problem in number theory.
Let $q\geq 2$ be an integer and $w$ be a finite string of elements in
$\{0,1,\dots,q-1\}$.
Then we define the sequence $\{e_q(w;n)\}_{n\geq1}$ to be the number of
occurrences of $w$ in the $q$-ary expansion of $n$.
The sequence $\{e_q(w;n)\}_{n\geq1}$ is called the pattern sequence
for $w$.
In this talk, we state the linear relations between pattern sequences.
Furthermore, by using Mahler's method, we obtain necessary and
sufficient conditions for algebraic dependence of generating functions
of pattern sequences.
Let $a$ be a fixed positive integer with $a\geq 2$, $p$ be an odd prime ($(a,p)=1$) and $D_a(p)$ be the (residual) order of $a$ in $Z/pZ^ \times$. The natural density $\Delta_a(q,0)$ of $p$ such that $q|D_a(p)$ ($q$: prime) is known. We obtain the natural density of $p$ such that $q|D_a(p)$ and $(b/p)=1$ under some slight restriction on $a$ and $b$, where $(b/p)$ is the Legendre symbol. Heuristically, it is half of $\Delta_a(q,0)$, but it is not true for some choices of $a$ and $b$. We also mention the results and methods of computer experiments.
$Date: 2013/11/15 04:47:06 $+ 9:00:00 (JST)