OSIDH and SiGamal : cryptosystems from supersingular elliptic curves (Theory and Applications of Supersingular Curves and Supersingular Abelian Varieties)

We introduce two cryptosystems, OSIDH and SiGamal, which use isogenies between supersingular elliptic curves over a finite field. And we consider computational problems on which these cryptosystems are based. In particular, we discuss a relation between these problems and problems to find the image of a point under a secret isogeny

A primality proving using elliptic curves with complex multiplication by imaginary quadratic fields of class number three

In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. For some technical reason, their framework can not be applied to an elliptic curve with complex multiplication by an imaginary quadratic field of class number greater than two. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. As an application, we give two special sequences of integers to which our method can be applied, and a computational result for the primality of these sequences

On the key generation in SQISign

SQISign is an isogeny-based signature scheme that has short keys and signatures and is expected to be a post-quantum scheme. Its security depends on the hardness of the problem to find an isogeny between given two elliptic curves over $\mathbb{F}_{p^2}$, where $p$ is a large prime. For efficiency reasons, a public key in SQISign is taken from a set of supersingular elliptic curves with a particular property. In this paper, we investigate the security related to public keys in SQISign. First, we show some properties of the set of public keys. Next, we show that a key generation procedure used in implementing SQISign could not generate all public keys and propose a modification for the procedure. In addition, we confirm the latter result through an experiment

Elliptic netを用いたpairing計算の高速化と並列計算

首都大学東京, 2017-03-25, 博士（理学）首都大学東

QFESTA: Efficient Algorithms and Parameters for FESTA using Quaternion Algebras

In 2023, Basso, Maino, and Pope proposed FESTA (Fast Encryption from Supersingular Torsion Attacks), an isogeny-based public-key encryption (PKE) protocol that uses the SIDH attack for decryption. In the same paper, they proposed a parameter for that protocol, but the parameter requires high-degree isogeny computations. In this paper, we introduce QFESTA (Quaternion Fast Encapsulation from Supersingular Torsion Attacks), a new variant of FESTA that works with better parameters using quaternion algebras and achieves IND-CCA security under QROM. To realize our protocol, we construct a new algorithm to compute an isogeny of non-smooth degree using quaternion algebra and the SIDH attack. Our protocol relies solely on $(2,2)$-isogeny and $3$-isogeny computations, promising a substantial reduction in computational costs. In addition, our protocol has significantly smaller data sizes for public keys and ciphertexts, approximately one-third the size of the original FESTA