Let [Formula: see text] be an integer such that has genus text], and let a field of characteristic or relatively prime to text]. In this paper, we explicitly classify the isogeny graphs all rational elliptic curves admit nontrivial over We achieve by introducing parameterized families defined which have following two properties for fixed text]: are isogenous there integers text]-invariants give...

Cryptosystems based on supersingular isogeny are a novel tool in post-quantum cryptography. One compelling characteristic is their concise keys and ciphertexts. However, the performance of computation currently worse than that other schemes. This primarily due to following factors. Firstly, underlying field quadratic extension finite field, resulting higher computational complexity. Secondly, s...

Article history: Received 15 August 2010 Revised 31 December 2010 Accepted 3 January 2011 Available online xxxx Communicated by David Goss MSC: primary 11G18, 11G09 secondary 14H40

The usage of elliptic curve cryptography in smart cards has been shown to be efficient although, when considering curves, one should take care about their vulnerability against the Zero-Value Point Attacks (ZVP). In this paper, we present a new procedure to find elliptic curves which are resistant against these attacks. This algorithm finds, in an efficient way, a secure curve by means of volca...

We present the first fault attack on cryptosystems based on supersingular isogenies. During the computation of the auxiliary points, the attack aims to change the base point to a random point on the curve via a fault injection. We will show that this would reveal the secret isogeny with one successful perturbation with high probability. We will exhibit the attack by placing it against signature...

Consider the abelian category Ck of commutative group schemes of finite type over a field k. By results of Serre and Oort, Ck has homological dimension 1 (resp. 2) if k is algebraically closed of characteristic 0 (resp. positive). In this article, we explore the abelian category of commutative algebraic groups up to isogeny, defined as the quotient of Ck by the full subcategory Fk of finite k-g...

Fix a prime number . Graphs of isogenies of degree a power of are well-understood for elliptic curves, but not for higher-dimensional abelian varieties. We study the case of absolutely simple ordinary abelian varieties over a finite field. We analyse graphs of so-called l-isogenies, resolving that, in arbitrary dimension, their structure is similar, but not identical, to the “volcanoes” occurri...

We propose an undeniable signature scheme based on elliptic curve isogenies, and prove its security under certain reasonable number-theoretic computational assumptions for which no efficient quantum algorithms are known. Our proposal represents only the second known quantum-resistant undeniable signature scheme, and the first such scheme secure under a number-theoretic complexity assumption.