Related papers: Ramanujan graphs in cryptography
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…
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.…
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…
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…
This paper contains a survey of supersingular isogeny graphs associated to supersingular elliptic curves and their various applications to cryptography. Within limitation of space, we attempt to address a broad audience and make this part…
Cryptographic hash functions from expander graphs were proposed by Charles, Goren, and Lauter in [CGL] based on the hardness of finding paths in the graph. In this paper, we propose a new candidate for a hash function based on the hardness…
Last year Takashima proposed a version of Charles, Goren and Lauter's hash function using Richelot isogenies, starting from a genus-2 curve that allows for all subsequent arithmetic to be performed over a quadratic finite field Fp2. In a…
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…
We consider finite graphs whose vertexes are supersingular elliptic curves, possibly with level structure, and edges are isogenies. They can be applied to the study of modular forms and to isogeny based cryptography. The main result of this…
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…
SIDH is a post-quantum key exchange algorithm based on the presumed difficulty of finding isogenies between supersingular elliptic curves. However, SIDH and related cryptosystems also reveal additional information: the restriction of a…
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…
Charles, Goren, and Lauter [J. Cryptology 22(1), 2009] explained how one can construct hash functions using expander graphs in which it is hard to find paths between specified vertices. The set of solutions to the classical Markoff equation…
We initiate the study of computational problems on elliptic curve isogeny graphs defined over RSA moduli. We conjecture that several variants of the neighbor-search problem over these graphs are hard, and provide a comprehensive list of…
In 2022, a prominent supersingular isogeny-based cryptographic scheme, namely SIDH, was compromised by a key recovery attack. However, this attack does not undermine the isogeny path problem, which remains central to the security of…
Building on work of Cai, F\"urer, and Immerman \cite{CFI92}, we show two hardness results for the Graph Isomorphism problem. First, we show that there are pairs of nonisomorphic $n$-vertex graphs $G$ and $H$ such that any sum-of-squares…
Hash functions map data of arbitrary length to data of predetermined length. Good hash functions are hard to predict, making them useful in cryptography. We are interested in the elliptic curve CGL hash function, which maps a bitstring to…
As quantum computing technology continues to advance, post-quantum cryptographic methods capable of resisting quantum attacks have emerged as a critical area of focus. Given the potential vulnerability of existing homomorphic encryption…
Starting from the one-way group action framework of Brassard and Yung (Crypto '90), we revisit building cryptography based on group actions. Several previous candidates for one-way group actions no longer stand, due to progress both on…
In their seminal paper, Lubotzky, Phillips and Sarnak (LPS) defined the notion of regular Ramanujan graphs and gave an explicit construction of infinite families of $(p+1)$-regular Ramanujan Cayley graphs, for infinitely many primes $p$. In…