Universality theorems of Selberg zeta functions for arithmetic groups2024 · Abstract We prove a universality theorem for the Selberg zeta function of subgroups of $\mathrm{SL}_2(\mathbb{Z})$ or co-compact arithmetic groups derived from quaternion algebras, in the strip $\{5/6 \lt \mathrm{Re}{s} \lt 1\}$, improving the range compared with a previous work by Drungilas–Garunkštis–Kačenas. We also obtain the same range for a joint universality theorem for congruence subgroups, which improves a result by Mishou.
Square integrals of the logarithmic derivatives of Selberg's zeta functions in the critical strip2021 · In our previous work (Y. Hashimoto, Selberg’s zeta function for the modular group in the critical strip, Math. Nachr. 294 (2021) 1899–1904, https://doi.org/10.1002/mana.202000268 ), we proposed an upper bound of the logarithmic derivative of Selberg’s zeta function for the modular group in the critical strip. This paper studies the growth of its square integral for the modular group, co-compact arithmetic groups derived from indefinite quaternion algebras and their subgroups.
Solving the problem of Blockwise Isomorphism of Polynomials with Circulant matrices2020 · <p>The problem of Isomorphism of Polynomials (IP problem) is known to be important to study the security of multivariate public key cryptosystems, one of the major candidates of post-quantum cryptography, against key recovery attacks. In these years, several schemes based on the IP problem itself or its generalization have been proposed. At PQCrypto 2020, Santoso introduced a generalization of the problem of Isomorphism of Polynomials, called the problem of Blockwise Isomorphism of Polynomials (BIP problem), and proposed a new Diffie-Hellman type encryption scheme based on this problem with Circulant matrices (BIPC problem). Quite recently, Ikematsu et al. proposed an attack called the linear stack attack to recover an equivalent key of Santoso's encryption scheme. While this attack reduced the security of the scheme, it does not contribute to solving the BIPC problem itself. In the present paper, we describe how to solve the BIPC problem directly by simplifying the BIPC problem due to the conjugation property of circulant matrices. In fact, we experimentally solved the BIPC problem with the parameter, which has 256 bit security by Santoso's security analysis and has 72.7bit security against the linear stack attack, by about 10 minutes.</p>
Key recovery attack on Hufu-UOV2019 · <p> The unbalanced oil and vinegar signature scheme (UOV) is a signature scheme whose public key is a set of quadratic polynomials over a finite field. This scheme has been considered to be secure and efficient enough under suitable parameter selections. However, its key size is relatively large, and then various arrangements of UOV with smaller keys have been proposed. Hufu-UOV proposed by Tao in 2019 is one of such variants of UOV, whose keys are generated by circulant and Toeplitz matrices. In the present paper, we study the security of Hufu-UOV and propose an attack on it. </p>
