Correlations of multiplicities in length spectra for congruence subgroups

Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup \Gam_0(n) and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group.Comment: 17 page

Arithmetic expressions of Selberg's zeta functions for congruence subgroups

In Sarnak's paper, it was proved that the Selberg zeta function for SL(2,Z) is expressed in terms of the fundamental units and the class numbers of the primitive indefinite binary quadratic forms. The aim of this paper is to obtain similar arithmetic expressions of the logarithmic derivatives of the Selberg zeta functions for congruence subgroups of SL(2,Z). As applications, we study the Brun-Titchmarsh type prime geodesic theorem, the asymptotic behavior of the sum of the class number.Comment: 12 page

Selberg's zeta function for the modular group in the critical strip

In the present paper, we study the growth of the Selberg zeta function for the modular group in the critical strip.Comment: 6 page

Universality theorems of the Selberg zeta functions for arithmetic groups

After Voronin proved the universality theorem of the Riemann zeta function in the 1970s, universality theorems have been proposed for various zeta and L-functions. Drungilas-Garunkstis-Kacenas' work at 2013 on the universality theorem of the Selberg zeta function for the modular group is one of them and is probably the first universality theorem of the zeta function of order greater than one. Recently, Mishou (2021) extended it by proving the joint universality theorem for the principal congruence subgroups. In the present paper, we further extend these works by proving the (joint) universality theorem for subgroups of the modular group and co-compact arithmetic groups derived from indefinite quaternion algebras, which is available in the region wider than the regions in the previous two works.Comment: 20 page

Weaknesses of cubic UOV and its variants

The unbalanced oil and vinegar signature scheme (UOV) is a signature scheme whose public key is a set of multivariate quadratic forms over a finite field. This -signature scheme has been considered to be secure and efficient enough under suitable parameter selections. However, the key size of UOV is relatively large and then reducing the key size of UOV is an important issue. Recently in Inscrypt 2015, a new variant of UOV called Cubic UOV was proposed, and in ICISC 2016, two variants of Cubic UOV called CSSv and SVSv were proposed. It has been claimed that these variants were more efficient than the original UOV and were secure enough. However, the security analyses of these schemes were not enough and they can be broken easily. In the present paper, we describe the weaknesses of these schemes

On the security of HMFEv

In this short report, we study the security of the new multivariate signature scheme HMFEv proposed at PQCrypto 2017