Schoof's algorithm and isogeny cycles
نویسندگان
چکیده
The heart of Schoof's algorithm for computing the cardinal-ity m of an elliptic curve over a nite eld is the computation of m modulo small primes`. Elkies and Atkin have designed practical improvements to the basic algorithm, that make use of \good" primes`. We show how to use powers of good primes in an eecient way. This is done by computing isogenies between curves over the ground eld. A new structure appears, called \isogeny cycle". We investigate some properties of this structure.
منابع مشابه
Parameters for Secure Elliptic Curve Cryptosystem - Improvements on Schoof's Algorithm
The security of elliptic curve cryptosystem depends on the choice of an elliptic curve on which cryptographic operations are performed. Schoof’s algorithm is used to define a secure elliptic curve, as it can compute the number of rational points on a randomly selected elliptic curve defined over a finite field. By realizing efficient combination of several improvements, such as Atkin-Elkies’s m...
متن کاملcient Implementation of Schoof ' s Algorithm
Schoof's algorithm is used to nd a secure elliptic curve for cryptosystems, as it can compute the number of rational points on a randomly selected elliptic curve de ned over a nite eld. By realizing e cient combination of several improvements, such as Atkin-Elkies's method, the isogeny cycles method, and trial search by match-and-sort techniques, we can count the number of rational points on an...
متن کاملEfficient Algorithms for Supersingular Isogeny Diffie-Hellman
We propose a new suite of algorithms that significantly improve the performance of supersingular isogeny Diffie-Hellman (SIDH) key exchange. Subsequently, we present a full-fledged implementation of SIDH that is geared towards the 128-bit quantum and 192bit classical security levels. Our library is the first constant-time SIDH implementation and is up to 2.9 times faster than the previous best ...
متن کاملFinding Good Random Elliptic Curves for Cryptosystems Deened over If 2 N
One of the main diiculties for implementing cryptographic schemes based on elliptic curves deened over nite elds is the necessary computation of the cardinality of these curves. In the case of nite elds IF2n, recent theoretical breakthroughs yield a signiicant speed up of the computations. Once described some of these ideas in the rst part of this paper, we show that our current implementation ...
متن کاملFPGA-SIDH: High-Performance Implementation of Supersingular Isogeny Diffie-Hellman Key-Exchange Protocol on FPGA
To the best of our knowledge, we present the first hardware implementation of isogeny-based cryptography available in the literature. Particularly, we present the first implementation of the supersingular isogeny Diffie-Hellman (SIDH) key exchange, which features quantum-resistance. We optimize this design for speed by creating a high throughput multiplier unit, taking advantage of parallelizat...
متن کامل