# Extending finite fields by Matan Prasma

- Channel: [ZuBerlin](https://streameth.org/zuberlin)
- Date: 2024-06-14
- Duration: 53:27
- Topics: cryptography, mathematics, blockchain, education, workshop
- Watch: https://streameth.org/watch/666c57f007f92b086c30922c
- Download: https://vod-cdn.lp-playback.studio/raw/jxf4iblf6wlsyor6526t4tcmtmqa/catalyst-vod-com/hls/dea54eawsmpwe45o/1080p0.mp4

## Description

Clip The video features a speaker from the Ethereum Foundation discussing the mathematics behind cryptography, particularly focusing on elliptic curves, finite fields, and pairings. The talk aims to provide an intuitive understanding of these concepts, with a special emphasis on extending finite fields. The speaker introduces basic polynomial equations with real coefficients and demonstrates how complex numbers allow for solutions when real numbers fail. The discussion then shifts to modular arithmetic with prime numbers, defining fields, and explaining the properties necessary for something to be considered a field.

The speaker goes on to draw parallels between polynomials over finite fields and integers, discussing concepts like prime polynomials, long division in polynomial rings, and the greatest common divisor (GCD) for polynomials. They also explore Bézout's lemma in the context of polynomials over finite fields.

Furthermore, the talk covers how to create fields from polynomials using modulo operations and establishes that these constructed fields are indeed fields according to their properties. The number of elements in such a field is determined by raising the size of the ground field to the degree of a given polynomial.

In conclusion, the speaker demonstrates an interesting trick that allows solving equations in larger fields when they can't be solved in smaller ones—analogous to extending real numbers to complex numbers—thus constructing Galois fields