This article was first published on Loopring Protocol - Medium
Loopring is creating a new “Learning Cryptography” series to educate the wider crypto community about this fascinating field. This series will begin from the basics, and work its way up to the advanced tools that make our scalable 3.0 DEX protocol — which utilises zero-knowledge proofs — possible.
For many developers like myself, understanding cryptography feels like a dark art/magic. It’s not that we find math hard, in fact many of us probably excelled in it in high school/college courses.
The problem lies with the fact that there’s no resource which balances the mathematics and presentation of ideas in an easy-to-understand manner. Facing this problem first hand, I decided to start this series to consolidate my learning, but also help various developers in their journey of understanding cryptography.
After reading through enough of these articles you should be able to understand and grasp more complicated cryptographic primitives in detail such as zero-knowledge proofs, threshold relay signatures and more.
Mathematical notation will gradually be introduced but pictures and repeated explanations will be present so you should hopefully be able to follow along!
Finite Fields, also known as Galois Fields, are cornerstones for understanding any cryptography. A field can be defined as a set of numbers that we can add, subtract, multiply and divide together and only ever end up with a result that exists in our set of numbers. This is particularly useful for crypto as we can deal with a limited set of extremely large numbers.
To have a finite field you need the following properties:
(the doughnut symbol ◦ denotes the remainder after multiplying/adding two elements)
- Closed — any operation performed with elements from the set returns an element contained in the original set.
- Associative — if you have (a ◦ b) ◦ c, it’s the same as a ◦ (b ◦ c)
- Identity — there exists a ...
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Loopring Protocol - Medium