In this paper, we study a balancing problem of ordinary binomial coefficients. The diophantine equation $$ {0\choose k}+{1\choose k}+\cdots+{x-1\choose k}={x+1\choose \ell}+\cdots+{y-1\choose \ell} $$ in positive integers $x$ and $y\ge x+2$ will be investigated, where $k$ and $\ell$ are given positive integers. The main results state effective and non-effective finiteness theorems on the subject matter. (joint work with Laszlo Szalay)
Let p be a prime number. We put ε = 1
if p ≡ 1 [3] and ε = -1 if p ≡ 2 [3].
Consider the recurrent
integer sequence (u_n)_{n∈N} defined by :
0 <u_0 < p is the initial term and :
u_{n+1} = u_n/3 if u_n ≡ 0 [3]
u_{n+1} = (p-εu_n)/3 if u_n ≡ 1 [3]
u_{n+1} = (p+εu_n)/3 if u_n ≡ 2 [3]
In this work, we show that this sequence
is connected to the hard discrete logarithm problem.
In particular, we prove that, under some favorable conditions, it
is possible to solve the discrete logarithm problem modulo p in the two following situations :
1. Number 3 is a primitive root modulo p.
2. The integer p is a safe prime: it has de form p = 2q+1, where
q is also prime.
Let s, M be two natural numbers. Take A, X which are sets of integers with | A | = s, | X| = M,. We put W = { a + x | a \in A, x \in X }. We call ( A, X ) a T-pair in case W makes a complete residue system modulo sM. Our problem is to list up ( A, X ). Nam ely (1) We construct T-pairs by aid of certain diagrams. It turns out T-pairs consist of standard S-pairs and curious HB-pairs. (2) All of T-pairs obtained by (1). To prove this assertion, we introduce ‘ Extract Method’.
In this talk, I will explain some relationship between dilogarithm functions and geometric invariants of knots, which appears in the study of the volume conjecture of knots.
Recently, the speaker wrote a program for computing Siegel modular forms of degree 2 and level 1 with Sage. In this talk, I will introduce the program by computing Fourier coefficients and eigenvalues of Hecke eigenforms of degree 2.
It is known that there are only finitely many elliptic curves having good reduction everywhere up to isomorphism over a number field by Shafarevich. As for the existence of elliptic curves having good reduction everywhere, we have many results. However, it is not easy to find all such curves over a given number field and we have only few fields over which all such curves are determined. We considered new method for determining elliptic curves with good reduction everywhere and carried out some calculation. In this talk, we introduce the method and results of the calculation.
Recently, Feng and Xiang (JCTA, 2012) found a new construction of skew Hadamard difference sets in elementary abelian groups. In this talk, we introduce a new invariant for equivalence of skew Hadamard difference sets, namely triple intersection numbers modulo a prime, and discuss inequivalence between Feng-Xiang skew Hadamard difference sets and the classical Paley difference sets. As a consequence, we show that their construction produces infinitely many skew Hadamard difference sets inequivalent to the Paley difference sets. Furthermore, we explain the importance of compuautions using computer in this study.
Poly-Euler numbers are introduced as an analogule of the poly-Bernoulli numbers due to Kaneko. In this talk, various properties of poly-Euler numbers, for instance, sign change, congruence relations, combinatorial interpretation and so on, are showed. An algorithm and inequality for the classical Euler numbers applied to the research are also presented.
Let m be a positive integer. Let us define {a_n} by a_1 = m, a_n+1 = a_n/2 (when a_n is even), and a_n+1 = 3a_n + 1 (when a_n is odd). It seems that, for any m, {a_n} ends by a_n = 1 (for some n). This conjecture is known as Collatz problem (or Kakutani problem). Concerning this problem, though many computational results are known, we hardly know about convergence or circulation. In this talk, we shall discuss Collatz-like problems using some functions on n instead of 3n+1
A subset X of a complex sphere is called a complex 3-code if the set of usual inner products of two distinct vectors in X has size 3. A complex 3-code naturally has the structure of a directed graph. One of main problems is to determine the largest size of 3-codes for a fixed dimension. When a directed graph is embedded to a complex sphere as a 3-code, we prove a necessary condition for a graph to be embedded as a largest 3-code, and introduce an algorithm to classify complex maximum 3-codes. The algorithm gives the classification for small dimensions.
One of the problem in design of experiments is that we have to determine a finite number of observation points and its weights to accurately grasp some characteristics of experimental objects. In this talk we give a geometric characterization of optimum experimental designs that consist of corner vectors associated with finite irreducible reflection groups.
It is known that the problem of finding the size of an optimal equi-difference conflict-avoiding code is concerned with the evaluation of the multiplicative order of a unit. In this talk, we will state the relation between the multiplicative order of 2 and equi-difference conflict-avoiding codes of odd length and weight 3, and show some properties of multiplicative order of a unit. Moreover, several explicit series of tight/optimal equi-difference conflict-avoiding codes of odd length and weight 3 are also given by revisiting several properties of multiplicative order of 2.
It is a fundamental and important problem in combinatorics and incidence geometry to find construction methods of orthogonal arrays. Fuji-hara and Kamimura (1993) proposed a construction method of orthogonal arrays with non-prime power number of symbols. In this talk, we improved this method and thereby experimentally give many new orthogonal arrays with much smaller runs and indices, compared with the arrays obtained from the earlier work. These new arrays also have a non-prime power number of symbols.
In this talk, we give a combinatorial proof for a theorem by Hilbert on the existence of Hilbert identities (Math. Ann., 1909), originaly designed as a key step in his solution of Waring's problem. Among many others, it has been believed that Ellison's proof (Amer. Math. Monthly, 1971) is the simplest one. However, in this talk, we point out a mistake in Ellison's proof, which we call "Ellison's error".
A Hadamard matrix gives rise to an association scheme of class $4$. In this talk, we generalize this to a balanced generalized weighing matrix. We also introduce the concept of mutually quasi-unbiased weighing matrices as a simultaneous generalization both for mutually unbiased bases and for mutually unbiased weighing matrices, and construct maximal examples of mutually quasi-unbiased weighing matrices.
We produce an efficient program package to compute the resultant matrix and its determinant for a given pair of multivariate polynomials on Magma. This package works much more faster than the Magma's built-in function "Resultant" for multivariate polynomials. We also explain some applications of this package, and especially, try some benchmark problem for computing general formula of the discriminant.
Matsumoto, Saito, and Matoba recently proposed the Walsh figure of merit (WAFOM), which is a computable criterion of quasi-Monte Carlo point sets constructed by digital nets. Matsumoto et al. also showed that the computational complexity is reduced for a special subclass of digital nets, and obtained concrete examples of low-WAFOM point sets by random search. In their framework, the number of points is fixed in advance. In this talk, we propose a random search algorithm for extensible low-WAFOM point sets based on general digital nets. For this, we also introduce a method using lookup tables to compute WAFOM faster.
Knapsack based cryptosystems using discrete logarithm problems for residue class fields of integer rings of number fields, which is ordinary called OTU2000, originally require quantum computers for key generation. For this, we shall propose some idea and refinement for practical use without quantum computers. We shall also discuss on the security problem which may arise by this refinement.
Modular forms and Galois representations attached to them have drawn many researchers' interest since the end of the 20thcentury. Theoretical investigations on them have been made rapidly. Now it becomes an important theme to compute them effectively and efficiently. In this talk, we give a brief survey on the computational aspects of modular forms.
In this talk, we report faster computation of "Round Robin GCD" which means GCDs of all paired combinations. Focusing on the target data (factors of relative class number of $p^{n+1}$-th cyclotomic fields) with deviation in distribution, we develop algorithm for fast computation of "Round Robin GCD". This work is carried out as part of joint work with Shoichi Nakajima (Gakushuin University) and Fumio Ichimura (Ibaraki University).
Several methods for generating elliptic curves usable in pairing-based cryptography enable to select a secure parameter according to any security level. However, whenever the parameter is changed, the definition polynomials of extension field and elliptic curves are also changed. This fact makes difficult to choose a best pairing algorithm. We present implementation-friendly pairing curves such that the definition polynomials of the elliptic curves and extension fields over which the curves are defined can be fixed, independently of security parameters. These elliptic curves and extension fields are obtained by the reduction of algebraic number fields and elliptic curves over algebraic number fields.
This paper suggests a pairing-based cryptography using super-elliptic curves in which a formula of canonical cohomology basis given the defining equation of the curve (Suzuki, 2013) and the Jacobi-variety representation by Harasawa-Suzuki (2000) that extended Mumford's for hyper-elliptic curves.
The security of pairing-based cryptography is based on the hardness of the discrete logarithm problem over GF(p^n). Joux et al. proposed the number field sieve over GF(p^n) at CRYPTO 2006 (JLSV06-NFS)as an extension of the number field sieve that can efficiently solve the discrete logarithm problem over GF(p) (JL03-NFS).In the sieving step of JL03-NFS we perform the sieving on 2 dimensions, but we have to deal with more than 2 dimensions in the case of JLSV06-NFS. In this paper, we present 3-dimensional lattice sieve as extension of 2-dimensional lattice sieve used by JLSV06-NFS.
The authors have proposed Cyclic Vector Multiplication Algorithm (CVMA) for extension field. CVMA consists of preparation step and evaluation step. The preparation step makes a certain table data and the evaluation step carries out a multiplication based on the table data. Since the table data depends on the fixed extension degree, the authors haven't considered its application to proper subfields. This research considers to apply CVMA for proper subfields without loss of efficiency.
$Date: 2013/12/19 08:40:51 $+ 9:00:00 (JST)