This post explores the group structure in the Tate pairing. I explain the terms embedding degree, torsion groups, twists and multiplicative group in the context of the Reduced Tate Pairing for BN Elliptic Curves.
Extension Fields Some time ago, I started a series on the algebraic structure behind ZK-Snarks. My focus was to explain why the group elements in the pairing libraries are represented the way they are. In the previous posts, I introduced groups, fields, elliptic curves and the general structure of pairings. In this post, I expand …
ZK Snarks and Their Algebraic Structure – Part 4 Read More »
My latest post continues exploring how to encode a verification problem inside a SNARK. This time, I explore the role of the Vanishing Polynomial when constructing a Quadratic Arithmetic Program to represent a set of constraints.
ZK-Snarks would not be possible without the use of bilinear pairings. In my series about their algebraic structure for beginners, I cover pairings in this post, after groups, fields and elliptic curves.
I give an explanation of what they are and how to operate with them, along with a simple example very famous in cryptography.
In a previous post I explained how one can express a provable function for a ZK-SNARK in a Rank-1 Constraint System. These constrains have to be transformed into a Quadratic Arithmetic Program before being usable to produce the Zero-Knowledge Proof. This post is the second post in a mini-series about Quadratic Arithmetic Programs, and explains …
Elliptic Curves Previously I started peeling away at the different components that make a cryptographic pairing. I talked about the general structure, and then gave a basic introduction to groups and fields. Today I pick up at that point and take you to the next step: elliptic curves. So let’s up the game a bit. …
ZK Snarks and Their Algebraic Structure – Part 2 Read More »
Currently at Artos, we are researching ZK SNARKs in depth. One of the tasks we’re handling is to convert a library written in Java (DIZK) to work with the same ZK SNARK scheme used by ZoKrates and libsnark, which are written in C++. Parts of this have already been detailed somewhere else in this blog. …
ZK-SNARKs are complex and hard to understand. For the most part, one can use them without knowing the mathematical details underlying them. Unless, that is, if you are developing your own ZK-SNARK solution.
This is the first in a series of posts about the internals of ZK-SNARKs and elliptic curves. Stay tuned for more.
In my experience, people smile when they hear the words Zero-Knowledge Proof . Maybe I did too, I don’t remember, that was a long time ago. But it is not uncommon to hear jibes like this “I’m an expert at Zero-Knowledge, I know nothing about it”.
In this post, I give a gentle introduction to the theme, so that you can have non-zero knowledge about ZKPs.
In the last post, I made an introduction to the concept of Zero Knowledge Proofs, and talked a bit about ZK-STARKs. Today, I’ll dive into the maths in Vitalik Buterin’s first post in a series about the subject. Vitalik does a good job of explaining them, but I feel the need to synthesise the information …
