Poor guy wants to learn some advanced math

How far are you with math? Before Christ? After Christ? 17th century? 19th century? Well, this poor guy would love to learn some 20th century abstract algebra and 21st century cryptography. Wish him luck. He will need it.


February 19, 2024

FFT in action

How the FFT helps speed up schemes like PLONK.

January 9, 2024

Schoof's algorithm

In 1985, René Schoof published an efficient algorithm for computing the number of points on elliptic curves over finite fields.

December 7, 2023

Montgomery modular multiplication

The computation of xy modulo p is much slower than usual multiplication, since it requires a division to know how many times p has to be subtracted from the product.

November 30, 2023

Roots in finite fields

How do we compute a square root of an element in a finite field of characteristics of some prime p?

October 16, 2023

FFT in finite fields

The finite field fast Fourier transform is incredibly useful in cryptography, in particular in modern ZKP schemes like PLONK.

September 27, 2023


Last time I wrote about the purpose of lookups and specifically about the Halo2 lookups. However, the first scheme for checking that the values of a committed polynomial are contained in some table was introduced in the paper "plookup: A simplified polynomial protocol for lookup tables".

June 9, 2023

Halo2 lookups

Lookups allow us to check whether a particular witness value is an element of some set. Such a check can be done using too but it’s much more expensive.

May 15, 2023

Permutation argument

There’s one very important idea in PLONK that I didn’t mention last time – the permutation argument.

April 17, 2023

Some ideas behind PLONK

PLONK stands for Permutations over Lagrange-bases for Oecumenical Noninteractive arguments of Knowledge. But do not mind the incomprehensible words – PLONK is an incredibly useful cryptographic protocol.

April 12, 2023

KZG commitments

Commitment schemes are used heavily in cryptography. In a nutshell, committing to some value means you cannot change this value later and this value is hidden to others. A Kate-Zaverucha-Goldberg commitment or simply Kate commitment allows commitment to a polynomial. And polynomial commitments are one of the building blocks of zk-rollups.

February 28, 2023

What is there to be studied?

The prominent British mathematician G. H. Hardy (1877-1947) said he had never done anything useful in his life. But don’t judge him. He did many wonderful things but is somehow, unfortunately, most famous for his discovery of Srinivasa Ramanujan (1887–1920).

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