Counting isomorphism classes of superspecial curves (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

A superspecial curve is a (non-singular) curve over a field of positive characteristic whose Jacobian variety is isomorphic to a product of supersingular elliptic curves over the algebraic closure. It is known that for given genus and characteristic, there exist only finitely many superspecial curves, up to isomorphism over an algebraically closed field. In this article, we give a brief survey on results of counting isomorphism classes of superspecial curves. In particular, this article summarizes some recent results in the case of genera four and five, obtained by the author and S. Harashita. We also survey results obtained in a joint work with Harashita and E. W. Howe, on the enumeration of superspecial curves in a certain class of non-hyperelliptic curves of genus four

Explicit construction of a plane sextic model for genus-five Howe curves, I

In the past several years, Howe curves have been studied actively in the field of algebraic curves over fields of positive characteristic. Here, a Howe curve is defined as the desingularization of the fiber product over a projective line of two hyperelliptic curves. In this paper, we construct an explicit plane sextic model for non-hyperelliptic Howe curves of genus five. We also determine singularities of our sextic model.Comment: Comments are welcome

Parametrizing generic curves of genus five and its application to finding curves with many rational points

In algebraic geometry, it is important to give good parametrizations of spaces of curves, theoretically and also practically. In particular, the case of non-hyperelliptic curves is the central issue. In this paper, we give a very effective parametrization of curves of genus $5$ which are neither hyperelliptic nor trigonal. After that, we construct an algorithm for a complete enumeration of generic curves of genus $5$ with many rational points, where "generic" here means non-hyperelliptic and non-trigonal with mild singularities of the associated sextic model which we propose. As an application, we execute an implementation on computer algebra system MAGMA of the algorithm for curves over the prime field of characteristic $3$.Comment: 16 page

A Variant of the XL Algorithm Using the Arithmetic over Polynomial Matrices (Computer Algebra : Foundations and Applications)

The title of this paper has been changed from the title of talk “Polynomial XL: A Variant of the XL Algorithm Using Macaulay Matrices over Polynomial Rings” at “Computer Algebra -Foundations and Applications”.Solving a system of multivariate polynomials is a classical but very important problem in many areas of mathematics and its applications, and in particular quadratic systems over finite fields play a major role in the multivariate public key cryptography. The XL algorithm is known to be one of the main approaches for solving a multivariate system, as well as Groebner basis approaches, and so far many variants of XL have been proposed. In this talk, we present a new variant of XL, which we name “Polynomial XL”, by using Macaulay matrices over polynomial rings

Attacks against search Poly-LWE

The Ring-LWE (RLWE) problem is expected to be a computationally-hard problem even with quantum algorithms. The Poly-LWE (PLWE) problem is closely related to the RLWE problem, and in practice a security base for various recently-proposed cryptosystems. In 2014, Eisentraeger et al. proposed attacks against the decision-variant of the PLWE problem (and in 2015, Elias et al. precisely described and extended their attacks to be applied for that of the RLWE problem). Their attacks against the decision-PLWE problem succeed with sufficiently high probability in polynomial time under certain assumptions, one of which is that the defining polynomial of the PLWE instance splits completely over the ground field. In this paper, we present polynomial-time attacks against the search-variant of the PLWE problem. Our attacks are viewed as search-case variants of the previous attacks, but can deal with more general cases where the defining polynomial of the PLWE problem does not split completely over the ground field

Polynomial XL: A Variant of the XL Algorithm Using Macaulay Matrices over Polynomial Rings

Solving a system of $m$ multivariate quadratic equations in $n$ variables over finite fields (the MQ problem) is one of the important problems in the theory of computer science. The XL algorithm (XL for short) is a major approach for solving the MQ problem with linearization over a coefficient field. Furthermore, the hybrid approach with XL (h-XL) is a variant of XL guessing some variables beforehand. In this paper, we present a variant of h-XL, which we call the \textit{polynomial XL (PXL)}. In PXL, the whole $n$ variables are divided into $k$ variables to be fixed and the remaining $n-k$ variables as ``main variables\u27\u27, and we generate a Macaulay matrix with respect to the $n-k$ main variables over a polynomial ring of the $k$ (sub-)variables. By eliminating some columns of the Macaulay matrix over the polynomial ring before guessing $k$ variables, the amount of manipulations required for each guessed value can be reduced. Our complexity analysis of PXL gives a new theoretical bound, and it indicates that PXL is efficient in theory on the random system with $n=m$, which is the case of general multivariate signatures. For example, on systems over ${\mathbb F}_{2^8}$ with $n=m=80$, the numbers of manipulations deduced from the theoretical bounds of the hybrid approaches with XL and Wiedemann XL and PXL with optimal $k$ are estimated as $2^{252}$, $2^{234}$, and $2^{220}$, respectively