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Finite Fields for Arithmetic Circuits

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This article is the third in a series. We present finite fields in the context of circuits for zero-knowledge proofs. The previous chapters are P vs NP and its Application to Zero Knowledge Proofs and Arithmetic Circuits. In the previous chapter on arithmetic circuits, we pointed out a limitation that we cannot encode the number 2/3 because it cannot be represented precisely using binary. We also pointed out that we didn’t explicitly have a way to handle overflow. Both of these issues can be handled seamlessly with a variant of arithmetic, which is popular in general cryptography, called finite fields. Given a prime number p, we can make a finite field with p elements by taking the set of integers {0, 1, 2, …, p-1} and define addition and multiplication to be done modulo p. We’ll start by limiting ourselves to fields where the number of elements is a prime. For example, if the prime number p is 7, then the elements in the finite field are {0, 1, 2, 3, 4, 5, 6}. Any number outside this

This article is the third in a series. We present finite fields in the context of circuits for zero-knowledge proofs. The previous chapters are P vs NP and its Application to Zero Knowledge Proofs and Arithmetic Circuits . In the previous chapter on arithmetic circuits, we pointed out a limitation that we cannot encode the number 2 / 3 2/3 2/3 because it cannot be represented precisely using binary. We also pointed out that we didn’t explicitly have a way to handle overflow. Both of these issues can be handled seamlessly with a variant of arithmetic, which is popular in general cryptography, c

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