Number Theory is the investigation of number fields, ie finite
extensions for the field of rational numbers. Those studies were
the result of investigations of irreducible polynomials, motivated
by Galois theory and the quest for Fermat's last theorem.
Zassenhaus later defined the four fundamental problems to be studied:
- the maximal order or ring of integers (as the canonical ring attached to the number field)
- the unit group of the ring of integers
- the class group (the multiplicative (ideal) structure)
- the Galois group
While algorithmic solutions to all four problems are known, they are
all under active research. In particular the problems related to the
class group and unit group are related to applications in cryptography.
In the talk I will sketch problems and techniques used to solve the
1st three problems, before discussing the problem of class groups and
their applications in more detail.
Theta functions are defined on principally polarised complex abelian varieties; they parameterise their space of moduli and yield an embedding into projective space. Their algebraic counterparts can play the same role for varieties defined over finite fields; with respect to cryptographic applications, abelian surfaces are particularly interesting.
I will report on a few results and algorithms obtained in my research group: fast addition formulae for Kummer surfaces suitable for cryptography; the computation of Shimura class fields of quartic CM fields, yielding surfaces with a known L-polynomial; and the computation of modular polynomials parameterising isogenies, maps between abelian surfaces.
(A triple of $(a,b,c)$ of positive integers is called a primitive Pythagorean triple if $a,b,c$ satisfy $a^2+b^2=c^2$ with $\gcd (a,b,c)=1$. In this talk, we firstly introduce several existing results on Je\'smanowicz' conjecture concerning a ternary Diophantine equation related to primitive Pythagorean triples. Next, we give the main result of this talk, which says that a quarter part of Je\'smanowicz' conjecture is regarded to be almost solved. Finally, we discuss the problem to improve the main theorem.)
In late 90s, Kaneko defined poly-Bernoulli numbers which are natural extensions of Bernoulli numbers. Thereafter, various relations among poly-Bernoulli numbers and mathematical objects were discovered one after another. In this talk, some of their properties and connections to other objects will be reviewed. Certain related works on Euler numbers will be also introduced.
In this talk, I am going to talk about linear relations among periods, such as multiple zeta values and periods of modular forms. I would like to share the basic tools for computing linear relations and explain what the dimension conjecture is and its application.
aIn this talk, we introduce the concept of the asymmetry for finite digraphs, as a natural analogue of the original concept for undirected graphs introduced by Erdo”s and Re’nyi in 1963. We prove an upper bound for the asymmetry of digraphs with n vertices. We show our bound is asymptotically best for sufficiently large orders of digraphs. We also investigate the asymmetry of countable digraphs. We also prove that countable random digraphs are almost surely symmetric, and isomorphic to the oriented random graph RO. We moreover show the cardinality of Aut(RO).
Delsarte gave an upper bound for the covering radius of a linear code based on the weight enumerator. When this bound is met for certain extremal binary doubly even self-dual codes, there is a singly even self-dual code whose shadow has an extremal property. The uniquely determined weight enumerator of the shadow turns out to have a negative coefficient in some cases, showing the impossibility of attaining the Delsarte bound.
Every projective plane P obtained from an abelian planar difference set of order n^2 has a Bear subplane P_0 and the substructure P \ P_0 is an (n^2 + n +1, n^2 ? n, n^2, 1)-DD (divisible design) D, which has the set C of n^2 + n + 1 point classes and admits an automorphism group (an SCT group) acting regularly on it. Conversely, such a DD admitting an SCT group can be extended to a projective plane of order n^2. Therefore, including non-abelian cases, we show that this gives new information concerning restriction of n on the existence of planar difference sets of order n^2 and (n^2 + n + 1, n^2 ? n, n^2, 1) RDSs.
An $n$-point subset $X$ of a $d$-dimensional Euclidean space $\mathbb{R}^d$ is called a $k$-distance set if exactly $k$ Euclidean distances occur between two distinct points in $X$. In this talk, I will introduce some classification problems of planar distance sets. In particular, we will discuss about distance sets on a circle. We show that if $k$ is small enough relative to $n$, then $n$-point $k$-distance set $X$ lies on a regular polygon. I also introduce an application of Kneser’s theorem. This work is based on a joint work with Koji Momihara.
Let $L_{mkl}\subset \mathbb{R}^{m+k+l}$ be the set of vectors which have $m$ of entries $-1$, $k$ of entries $0$, and $l$ of entries $1$. In this talk, we show the classification of the largest subset of $L_{1k2}$ whose diameter is smaller than that of $L_{1k2}$. From this result, we can classify the largest $4$-distance sets containing the Euclidean representation of the Johnson scheme $J(9,4)$. This was an open problem in Bannai, Sato, and Shigezumi (2012).
When the author was developing the crystallographic powder indexing software Conograph, some needs for new types of algebraic algorithms arose. In particular, “error-stable Bravais lattice determination” and “distribution rules of space-group systematic absences” are introduced in the talk, from the viewpoints of both mathematics and crystallography.
We show that the solutions of a polynomial in binary variables are obtained from the eigenvectors of the linear representation of a special residue class ring. In particular, the trace equals the summation of a polynomial in binary variables, which can be computed by substituting the midpoint for its residue class. We expect that this formula give an efficient method for multiple integrals.
In the past year, Miller's algorithm has been used for computing pairings. In 2007, Stange defined elliptic nets and proposed an alternative method for computing pairings based on them. We give an explicit formula for computing the optimal ate pairing over the BN curve via the elliptic net associated to the twist curve and construct algorithms to parallelize a computation of this elliptic net.
Pairing-based cryptographic schemes require so-called pairing-friendly elliptic curves which have special properties. The families of pairing-friendly elliptic curves which are generated by polynomials are called complete families. Although a complete family with the rho-value 1 is an ideal case, there is the only known example which is given by Barreto and Naehrig. We prove that many complete families are non-ideal.
The encryption scheme NTRU is designed over a polynomial ring. Basically, if the ring is changed to any other ring, NTRU-like cryptosystem is constructible. In this talk, we consider variants of NTRU using various rings, Moreover, we analyze and compare the security of these schemes using the irreducible decomposition of the rings.
LMFDB (the database of L-functions, Modular forms, and related objects) is an extensive database (project) in number theory. In this talk, we give a report about some contribution to LMFDB, and collaboration with Sage Math project.
In this talk, method of calculation of relative class number, calculated range, future problem and so on, are showed. Relative class number of cyclotomic field is defined by dividing class number of cyclotomic field by class number of maximal subfueld of cyclotomic field. The relative class number has emerged is the 19th century. Unresolved issues such as prime divisor and magnitude of value and so on, are left many. Calculation method for relative class number are mainly classified into those using determinant, finite field, and polynomial. We introduce for this to calculation of relative class number, discussed with the ingenuity of the speakers themselves and collaborators. This work is based on a joint work with Shoichi Nakajima and Fumio Ichimura.
The research of discrete logarithm problems over finite fields of small characteristic is one of important researches in cryptography, since the safeties of several useful public key cryptosystems depend on the difficulty of solving such discrete logarithm problems. Many innovative improvements of algorithms to solve such discrete logarithm problems are recently proposed. This talk presents a survey of them.
It is important to compute the dimensions of the cohomology groups of coherent sheaves on projective schemes over a field, because the dimensions are used for computing some geometric invariant. After J.-P. Serre proved the possibility of the computation, some algorithms for the computation have been proposed. In this talk, we describe such an algorithm based on Groebner bases for free modules, and give a demonstration by using a function which we implemented in the computer algebra system Magma. After that, we compare our function with the built-in function in Magma, and give an observation for higher-speed computation.
$Date: 2015/11/24 04:43:16 $+ 9:00:00 (JST)