arXiv subjects

Tsuyoshi Takagi

Publications and source records attributed to Tsuyoshi Takagi.

6 recordsLinked to original sources

arXiv2026

An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences

We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties conjectured to hold for generic sequences-specifically, that their leading monomial ideals are weakly reverse lexicographic and that their Hilbert series follow a known closed-form expression. The algorithm incrementally constructs the set of leading monomials degree by degree by comparing Hilbert functions of monomial ideals with the expected Hilbert series of the input ideal. To enhance computational efficiency, we introduce several optimization techniques that progressively narrow the search space and reduce the number of divisibility checks required at each step. We also refine the loop termination condition using degree bounds, thereby avoiding unnecessary recomputation of Hilbert series. Experimental results confirm that the proposed method substantially reduces both computation time and memory usage compared to conventional Groebner basis computations for computing the leading monomials of a minimal Groebner basis of generic sequences.

cs.SC↗

arXiv2025

Enumeration Algorithm for Genus-4 Superspecial Hyperelliptic Curves with Automorphism Group $Q_8$

In this paper, we propose an algorithm to enumerate genus-4 superspecial hyperelliptic curves whose automorphism groups isomorphic to the quaternion group. By implementing this algorithm with Magma, we successfully obtain the number of isomorphism classes of such curves in every characteristic $7 \leq p < 10000$. Interestingly, the experimental results lead us to the conjecture that there exist exactly $[p/48]$ isomorphism classes of such curves if $p \equiv 1,7 \pmod{8}$, whereas such curves exist if $p \equiv 3,5 \pmod{8}$

math.AG↗

arXiv2022

Efficient search for superspecial hyperelliptic curves of genus four with automorphism group containing $\mathbb{Z}_6$

In arithmetic and algebraic geometry, superspecial (s.sp.\ for short) curves are one of the most important objects to be studied, with applications to cryptography and coding theory. If $g \geq 4$, it is not even known whether there exists such a curve of genus $g$ in general characteristic $p > 0$, and in the case of $g=4$, several computational approaches to search for those curves have been proposed. In the genus-$4$ hyperelliptic case, Kudo-Harashita proposed a generic algorithm to enumerate all s.sp.\ curves, and recently Ohashi-Kudo-Harashita presented an algorithm specific to the case where automorphism group contains the Klein 4-group. In this paper, we propose an algorithm with complexity $\tilde{O}(p^4)$ in theory but $\tilde{O}(p^3)$ in practice to enumerate s.sp.\ hyperelliptic curves of genus 4 with automorphism group containing the cyclic group of order $6$. By executing the algorithm over Magma, we enumerate those curves for $p$ up to $1000$. We also succeeded in finding a s.sp.\ hyperelliptic curve of genus $4$ in every $p$ with $p \equiv 2 \pmod{3}$. As a theoretical result, we classify hyperelliptic curves of genus $4$ in terms of automorphism groups in the appendix.

math.AG↗

arXiv2022

A physical study of the LLL algorithm

This paper presents a study of the LLL algorithm from the perspective of statistical physics. Based on our experimental and theoretical results, we suggest that interpreting LLL as a sandpile model may help understand much of its mysterious behavior. In the language of physics, our work presents evidence that LLL and certain 1-d sandpile models with simpler toppling rules belong to the same universality class. This paper consists of three parts. First, we introduce sandpile models whose statistics imitate those of LLL with compelling accuracy, which leads to the idea that there must exist a meaningful connection between the two. Indeed, on those sandpile models, we are able to prove the analogues of some of the most desired statements for LLL, such as the existence of the gap between the theoretical and the experimental RHF bounds. Furthermore, we test the formulas from the finite-size scaling theory (FSS) against the LLL algorithm itself, and find that they are in excellent agreement. This in particular explains and refines the geometric series assumption (GSA), and allows one to extrapolate various quantities of interest to the dimension limit. In particular, we predict the empirical average RHF converges to $\approx 1.02265$ as dimension goes to infinity.

cond-mat.stat-mech↗

arXiv2020

LLL and stochastic sandpile models

Theaimofthepresentpaperistosuggestthatstatisticalphysicsprovides the correct language to understand the practical behavior of the LLL algorithm, most of which are left unexplained to this day. To this end, we propose sandpile models that imitate LLL with compelling accuracy, and prove for these models some of the most desired statements regarding LLL. We also formulate a few conjectures that formally capture our heuristics and would serve as milestones for further development of the theory.

math.NT↗

arXiv2014

Some properties of $τ$-adic expansions on hyperelliptic Koblitz curves

This paper explores two techniques on a family of hyperelliptic curves that have been proposed to accelerate computation of scalar multiplication for hyperelliptic curve cryptosystems. In elliptic curve cryptosystems, it is known that Koblitz curves admit fast scalar multiplication, namely, the $τ$-adic non-adjacent form ($τ$-NAF). It is shown that the $τ$-NAF has the three properties: (1) existence, (2) uniqueness, and (3) minimality of the Hamming weight. These properties are not only of intrinsic mathematical interest, but also desirable in some cryptographic applications. On the other hand, G{ü}nther, Lange, and Stein have proposed two generalizations of $τ$-NAF for a family of hyperelliptic curves, called \emph{hyperelliptic Koblitz curves}. However, to our knowledge, it is not known whether the three properties are true or not. We provide an answer to the question. Our investigation shows that the first one has only the existence and the second one has the existence and uniqueness. Furthermore, we shall prove that there exist 16 digit sets so that one can achieve the second one.

math.NT↗