Related papers: Hard isogeny problems over RSA moduli and groups w…

An important open problem in supersingular isogeny-based cryptography is to produce, without a trusted authority, concrete examples of "hard supersingular curves" that is, equations for supersingular curves for which computing the…

Given two elliptic curves over a finite field having the same cardinality and endomorphism ring, it is known that the curves admit an isogeny between them, but finding such an isogeny is believed to be computationally difficult. The fastest…

An isogeny between elliptic curves is an algebraic morphism which is a group homomorphism. Many applications in cryptography require evaluating large degree isogenies between elliptic curves efficiently. For ordinary curves of the same…

In this paper, we prove that the supersingular isogeny problem (Isogeny), endomorphism ring problem (EndRing) and maximal order problem (MaxOrder) are equivalent under probabilistic polynomial time reductions, unconditionally. Isogeny-based…

In this paper, we study the problem of sampling random supersingular elliptic curves with unknown endomorphism rings. This problem has recently gained considerable attention as many isogeny-based cryptographic protocols require such…

In this paper, we study isogeny graphs of supersingular elliptic curves. Supersingular isogeny graphs were introduced as a hard problem into cryptography by Charles, Goren, and Lauter for the construction of cryptographic hash functions…

Polycyclic groups are natural generalizations of cyclic groups but with more complicated algorithmic properties. They are finitely presented and the word, conjugacy, and isomorphism decision problems are all solvable in these groups.…

We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the…

We prove that the path-finding problem in $\ell$-isogeny graphs and the endomorphism ring problem for supersingular elliptic curves are equivalent under reductions of polynomial expected time, assuming the generalised Riemann hypothesis.…

Given a cryptographic group action, we show that the Group Action Inverse Problem (GAIP) and other related problems cannot be NP-hard unless the Polynomial Hierarchy collapses. We show this via random self-reductions and the design of…

We employ methods from homotopy theory to define new obstructions to solutions of embedding problems. By using these novel obstructions we study embedding problems with non-solvable kernel. We apply these obstructions to study the…

We investigate the isogeny graphs of supersingular elliptic curves over $\mathbb{F}_{p^2}$ equipped with a $d$-isogeny to their Galois conjugate. These curves are interesting because they are, in a sense, a generalization of curves defined…

We present e cient algorithms for computing isogenies between hyperelliptic curves, leveraging higher genus curves to enhance cryptographic protocols in the post-quantum context. Our algorithms reduce the computational complexity of isogeny…

Supersingular elliptic curve isogeny graphs underlie isogeny-based cryptography. For isogenies of a single prime degree $\ell$, their structure has been investigated graph-theoretically. We generalise the notion of $\ell$-isogeny graphs to…

Onion routing provides anonymity by layering encryption so that no relay can link sender to destination. A quantum analogue faces a core obstacle: layered quantum encryption generally requires symmetric encryption schemes, whereas…

Most common public key cryptosystems and public key exchange protocols presently in use, such as the RSA algorithm, Diffie-Hellman, and elliptic curve methods are number theory based and hence depend on the structure of abelian groups. The…

We consider the isomorphism problem for hypergraphs taking as input two hypergraphs over the same set of vertices $V$ and a permutation group $\Gamma$ over domain $V$, and asking whether there is a permutation $\gamma \in \Gamma$ that…

Fix an ordinary abelian variety defined over a finite field. The ideal class group of its endomorphism ring acts freely on the set of isogenous varieties with same endomorphism ring, by complex multiplication. Any subgroup of the class…

We give an overview of the Hidden Subgroup Problem (HSP) as of July 2010, including new results discovered since the survey of arXiv:quant-ph/0411037v1. We recall how the problem provides a framework for efficient quantum algorithms and…

An isogeny graph is a graph whose vertices are principally polarized abelian varieties and whose edges are isogenies between these varieties. In his thesis, Kohel described the structure of isogeny graphs for elliptic curves and showed that…