We outline PARI programs which assist with various algorithms related to descent via isogeny on elliptic curves. We describe, in this context, variations of standard inequalities which aid the computation of members of the Tate-Shafarevich group. We apply these techniques to several examples: in one case we use descent via 9-isogeny to determine the rank of an elliptic curve; in another case we...

Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These resul...

We present the first constant-round, tree-based, group key exchange protocol based on SIDH with logarithmic-order communication and memory complexity, where only previous isogeny-based exchange, SIBD, has linear-order complexity. call our supersingular isogeny tree-based (SIT). show that satisfies post-quantum security through a reduction to decisional Diffie–Hellman (SSDDH) problem in model of...

We present a new algorithm to compute the classical modular polynomial Φl in the rings Z[X,Y ] and (Z/mZ)[X, Y ], for a prime l and any positive integer m. Our approach uses the graph of l-isogenies to efficiently compute Φl mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected run...

In this expository note, we describe an arithmetic pairing associated to an isogeny between Abelian varieties over a finite field. We show that it generalises the Frey–Rück pairing, thereby giving a short proof of the perfectness of the latter.