# What basic knowledge is required to understand SIKE?

I'm interested in learning about Supersingular Isogeny based Key Encapsulation mechanism. Currently, I only know all the basic knowledge about how standard Elliptic Curve Cryptography, works, using Weierstrass equations. This includes the the mathematical group structure on how the Elliptic curve points are defined, and operations like point doubling and addition. I also learned how to implement it in C , with parameters of cryptographic size.

Since, SIKE is dependent on Elliptic curves, and mapping between curves, I'm guessing that my basic knowledge of how Classical Elliptic Curve Crypto works, would provide me (a beginner) some vantage point to begin understand how SIKE takes off from the ground up.

I realise, that the security of SIKE is not dependent on the classical Discrete Logarithm problem.

I cannot find online material, that presents SIKE and the mathematical basis required for it for a beginner. Even, if that's not the case, I'm willing to learn the math required from the ground up.

Every, help will be greatly appreciated! Thankyou, everyone in advance!

## 1 Answer

Craig Costello has tried writing precisely what you're interested in, see SIKE for Beginners. It refers to numerous other surveys for isogeny-based crypto which may be useful (lecture notes by De Feo, and surveys by Galbraith-Vercauteren and Smith).

Galbraith also has a book on the math behind crypto. Chapter 25 deals with isogenies, so may be useful for mathematical background (but does not discuss SIKE).

• Thankyou, As a beginner to Post Quantum crypto, I dont have in depth knowledge of other PQ cryptosystems. I chose to learn SIKE because of its roots in Elliptic curve over finite fields. Do you think, that SIKE will be more cryptographically secure than the other candidates based on Lattice based crypto? – Vivekanand V Jul 27 '20 at 6:42
• @VivekanandV It's really premature to say. Lattice-based crypto (say NTRU-based) is older by roughly a decade and has resisted cryptanalysis, while LWE-based lattice crypto has connections to hard problems in coding theory, and seems fairly safe. The real question is whether RLWE (or other "algebraically structured" LWE) is safe --- without this LWE-based schemes are too inefficient. So far this seems to be the case (and there are some concrete things like the middle product LWE reductions which make this seem more likely to be true), but some attack on RLWE which makes use of its ... – Mark Jul 27 '20 at 15:08
• algebraic structure is entirely possible, so there could definitely be a world where LWE is secure, RLWE isn't (so LWE is too inefficient), and therefore SIKE is preferable. Regardless either Lattices or Isogenies seem like fine areas to research, and if you have the mathematical maturity for one it's likely that you can "pivot" to the other if any issues arise --- so I would choose based off of personal interest (for example, I always liked "linear algebraic intuition", so lattices pair fairly well with that). – Mark Jul 27 '20 at 15:10