Trusted Setup with Isogenies
Trusted parties are fundamental in setting up secure communication among parties. For instance, a trusted setup is needed when establishing a trusted relationship between users and certain public information in a public-key infrastructure (PKI) for public-key encryption and signature schemes.
The risk with placing trust on a third party can be ...
Pseudorandom Functions from Isogenies
In this blogpost we will assume knowledge of CSIDH (see previous blogpost). Basic knowledge of pseudorandom functions will be useful.
What is a Pseudorandom Function?
Intuitively, a pseudorandom function (PRF) is a function that “looks like” a random function. We now define it more formally.
Firstly, we let $F: ${$0,1$}$^* \times ${$0,1$}$^*...
CSIDH
For this blogpost we will assume knowledge of
Preview blogpost on Isogenies for Cryptography
Optional: Previous blogpost on SIDH
We will also need a basic understanding of ideal class groups.
Ideal Class Groups
We briefly introduce ideal class groups in the context needed for CSIDH. For a more general treatment and prec...
Asiacrypt 2020
This year at Asiacrypt a total of 7 papers on isogeny-based cryptography were presented. For those who couldn’t attend, I’ve briefly described each paper. I also give links to the papers and corresponding talks (on YouTube) for more details.
SQISign: Compact Post Quantum signatures from Quaternions and Isogenies
[Paper]
Winning the best paper ...
SIDH
SIDH
Stolbunov proposed a Diffie-Hellman type scheme based on the difficulty of computing isogenies between ordinary elliptic curves, with the aim of obtaining quantum-resistant cryptographic protocols 1. The fastest known classical probabilitic algorithm for solving this problem is an algorithm of Galbraith and Stolbunov 2, which is exponentia...
Vélu's Formulas for SIDH
In this blog post I will assume knowledge on:
Elliptic curves (over finite fields)
A previous blogpost on isogenies for cryptography
Polynomials
Suppose we have an elliptic curve $E_1$ over a finite field $\mathbb{F}_{p^2}$. Given the kernel of an isogeny (a finite subgroup of the group of points on $E_1$), Vélu’s formulas give us a wa...
Isogenies for Cryptography
On July 22, 2020, the Round 3 finalists for the NIST Post-quantum Cryptography Standardization effort were announced. One of the alternate candidates for public-key encryption and key-establishement algorithms is SIKE, a key encapsulation mechanism (KEM) based on isogenies. A non-specialist wanting a basic understanding of the schemes may find t...