The square of a k-th root of unity is a k/2-th root of unity If we take the set of $k$-th roots of unity (with $k$ even) and square each element, the resulting set will be a set of half the size. The...

Evaluating multivalued functions by square root expansion In the previous chapter on Image Preservation of Multivalued Functions we saw that instead of evaluating $f(x)$ on the $k$-th roots of unity,...

Square Roots of Roots of Unity The square root of a number $x$ is $y$ such that $y^2=x$. When $x$ is of the form $x^m$ and $m$ is even, then the square root is easy to compute: it’s simply $x^{m/2}$....

The Image Preservation Theorem for Multivalued Functions We’ll start this chapter on an unusual note — the NTT algorithm is quite simple and can be implemented in less than 20 lines of code. However,...

Vandermonde Matrices A Vandermonde matrix is a matrix that converts a polynomial from its coefficient representation into its value representation at a set of points. For a polynomial $f(x) = a_0...

Elliptic Curve Point Addition This article describes how elliptic curve addition works over real numbers. Cryptography uses elliptic curves over finite fields, but elliptic curves are easier to...

The Fundamental Theorem of Finite Cyclic Groups The Fundamental Theorem of Cyclic Groups provides guarantees about the existence of cyclic subgroups within a cyclic group. In the context of the...

Solidity Signed Integer Solidity signed integers enable using negative numbers in a smart contract. This article documents how they are used at the EVM level. Basic familiarity with the EVM and...

Smart contract creation cost Smart contract creation cost can be anywhere from \$10 to \$2,000 assuming ETH costs between \$1,500 to \$2,000. The biggest factors are 1) Ethereum price, 2) the size of...

Smart Contract Security !smart contract security This article serves as a mini course on smart contract security and provides an extensive list of the issues and vulnerabilities that tend to recur in...