An Approach to the Construction of a Recursive Argument of Polynomial Evaluation in the Discrete Log Setting

Succinct Non-interactive Arguments of Knowledge (SNARks) are receiving a lot of attention as a core privacy-enhancing technology for blockchain applications. Polynomial commitment schemes are important building blocks for the construction of SNARks. Polynomial commitment schemes enable the prover to commit to a secret polynomial of the prover and convince the verifier that the evaluation of the committed polynomial is correct at a public point later. Bünz et al. recently presented a novel polynomial commitment scheme with no trusted setup in Eurocrypt’20. To provide a transparent setup, their scheme is built over an ideal class group of imaginary quadratic fields (or briefly, class group). However, cryptographic assumptions on a class group are relatively new and have, thus far, not been well-analyzed. In this paper, we study an approach to transpose Bünz et al.’s techniques in the discrete log setting because the discrete log setting brings a significant improvement in efficiency and security compared to class groups. We show that the transposition to the discrete log setting can be obtained by employing a proof system for the equality of discrete logarithms over multiple bases. Theoretical analysis shows that the transposition preserves security requirements for a polynomial commitment scheme.

Galois covers of ℙ1 and number fields with large class groups

For each finite subgroup [Formula: see text] of [Formula: see text], and for each integer [Formula: see text] coprime to [Formula: see text], we construct explicitly infinitely many Galois extensions of [Formula: see text] with group [Formula: see text] and whose ideal class group has [Formula: see text]-rank at least [Formula: see text]. This gives new [Formula: see text]-rank records for class groups of number fields.

A proof of the conjectured run time of the Hafner-McCurley class group algorithm

<p style='text-indent:20px;'>We present a proof under a generalization of the Riemann Hypothesis that the class group algorithm of Hafner and McCurley runs in expected time <inline-formula><tex-math id="M1">\begin{document}$ e^{\left(3/\sqrt{8}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ -d $\end{document}</tex-math></inline-formula> is the discriminant of the input imaginary quadratic order. In the original paper, an expected run time of <inline-formula><tex-math id="M3">\begin{document}$ e^{\left(\sqrt{2}+o(1)\right)\sqrt{\log d\log\log d}} $\end{document}</tex-math></inline-formula> was proven, and better bounds were conjectured. To achieve a proven result, we rely on a mild modification of the original algorithm, and on recent results on the properties of the Cayley graph of the ideal class group.</p>

On the compositum of orthogonal cyclic fields of the same odd prime degree

The aim of this paper is to study circular units in the compositum K of t cyclic extensions of ${\mathbb {Q}}$ ( $t\ge 2$ ) of the same odd prime degree $\ell $ . If these fields are pairwise arithmetically orthogonal and the number s of primes ramifying in $K/{\mathbb {Q}}$ is larger than $t,$ then a nontrivial root $\varepsilon $ of the top generator $\eta $ of the group of circular units of K is constructed. This explicit unit $\varepsilon $ is used to define an enlarged group of circular units of K, to show that $\ell ^{(s-t)\ell ^{t-1}}$ divides the class number of K, and to prove an annihilation statement for the ideal class group of K.

On minus quotients of ideal class groups of cyclotomic fields

Let [Formula: see text] be the minus quotient of the ideal class group of the [Formula: see text]th cyclotomic field. In this paper, first, we show that each finite abelian group appears as a subgroup of [Formula: see text] for some [Formula: see text]. Second, we show that, for all pairs of integers [Formula: see text] and [Formula: see text] with [Formula: see text], the kernel of the lifting map [Formula: see text] is contained in the [Formula: see text]-torsion [Formula: see text] of [Formula: see text]. Such an evaluation of the exponent is an individuality of cyclotomic fields.