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Mathematics of Isogeny Based Cryptography

For those with a mathematical background, these lecture notes by

@luca_defeo

seem to be the best starting place to understand how cryptographic primitives are being created from isogeny based problems Mathematics of Isogeny Based Cryptography Luca De Feo

arxiv.org

These lectures notes were written for a summer school on Mathematics for post-quantum cryptography in Thiès, Senegal. They try to provide a guide for Masters' students to get through the vast...

For those more interested in an introduction / in-depth discussion of the SIKE protocol, I recommend this paper by Craig Costello. Supersingular isogeny key exchange for beginners eprint.iacr.org/2019/1321.pdf

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If you learn better from talks rather than papers, Craig also gave a webinar on this same paper which can be watched for free via

@MSFTResearch

's site Post-quantum cryptography: Supersingular isogenies for beginners

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Silverman's AEC is often cited as a great introduction for elliptic curves, but if you're like me and learning the maths as you learn your crypto, I really recommend Silverman and Tate's book "Rational points on elliptic curves" which sets a great groundwork for beginners

If you follow CryptoHack, chances are you like to learn by pulling things apart. This is an incredible review of some attacks on isogeny-based schemes On the Security of Supersingular Isogeny Cryptosystems eprint.iacr.org/2016/859.pdf

More recently, I've really enjoyed reading papers on torsion point attacks. Maybe one place to start is Petite's paper from 2017 eprint.iacr.org/2017/571.pdf

More recently, Lorenz Panny (who I know of because of HXP) co-authored an improvement on this paper with de Quehen, Kutas, Leonardi, Martindale, Petit and Stange. It's a clear and beautiful paper Improved torsion point attacks on SIDH variants eprint.iacr.org/2020/633

If you're anything like me, you need to learn a lot of these things more than once before it settles into place. I'm excited about the Isogeny School starting this summer, hosted by

@chloelono

and Christophe Petit. isogenyschool2020.co.uk

Today

and his co-authors had this pre-print looking at using torsion points to break security assumptions from a resent OPRF proposal put forward by Boneh et al. Read the first half during lunch and felt it was particularly clear and well written

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asanso.eth

@asanso

· May 28, 2021

"Cryptanalysis of an oblivious PRF fromsupersingular isogenies" eprint.iacr.org/2021/706 Joint work with @andreavbasso @merzsp @kutasp and Christophe Petit . Poc code in github.com/isogenists/iso

As a maths break, this review of 10 years of SIKE by Costello was a nice historical overview, which compared SIKE with other post-quantum proposals. We read it together in our discord, I suggest reading it eprint.iacr.org/2021/543.pdf

Jun 1, 2021

Really enjoyed reading

@mariascrs

’ blog, which has some really cool resources and external links. Found it while researching Vélu’s formula :) mariascrs.github.io

Replying to

Feeling motivated to put my money where my mouth is and make a little Isogeny section on the site...

Replying to

Hi CryptoHack, I'm curious about Isogeny-Based PQC, it's well known that Isogeny is slow in terms of performance, what is attractive about Isogeny? I see Isogeny is a very active research field.