## Lagrange interpolation

Recall from our basics discussion that a polynomial $\phi$ of degree $d$ is a vector of $d+1$ coefficients:
\begin{align}
\phi &= [\phi_0, \phi_1, \phi_2, \dots, \phi_d]
\end{align}
How to compute a polynomial’s coefficients from a bunch of its evaluations
Given $n$ pairs $(x_i, y_i)_{i\in[n]}$, one can compute or interpolate a degree...

## Feist-Khovratovich technique for computing KZG proofs fast

tl;dr: Given a polynomial $f(X)$ of degree $m$, can we compute all $n$ KZG proofs for $f(\omega^k), k\in[0,n-1]$ in $O(n\log{n})$ time, where $\omega$ is a primitive $n$th root of unity?
Dankrad Feist and Dmitry Khovratovich[^FK20] give a resounding ‘yes!’

## Basic Group Theory

Multiplicative inverses modulo $m$
The multiplicative group of integers modulo $m$ is defined as:
\begin{align}
\Z_m^* = \{a\ |\ \gcd(a,m) = 1\}
\end{align}
But why?
This is because Euler’s theorem says that:
\begin{align}
\gcd(a,m) = 1\Rightarrow a^{\phi(m)} = 1
\end{align}
This in turn, implies that every element in $\Z_m^*$ has an invers...

## What is a Merkle tree?

tl;dr: In this post, we demystify Merkle trees for beginners.
We give simple, illustrated explanations, we detail applications they are used in and reason about their security formally.
For more details, see this full post on Decentralized Thoughts.

## Authenticated Dictionaries with Cross-Incremental Proof (Dis)aggregation

tl;dr: We build an authenticated dictionary (AD) from Catalano Fiore vector commitments that has constant-sized, aggregatable proofs and supports a stronger notion of cross-incremental proof disaggregation.
Our AD could be used for stateless validation in cryptocurrencies with smart contract execution.
In a future post, we will extend this AD wi...

## RSA Accumulators

An RSA accumulator is an authenticated set built from cryptographic assumptions in hidden-order groups such as $\mathbb{Z}_N^*$.
RSA accumulators enable a prover, who stores the full set, to convince any verifier, who only stores a succinct digest of the set, of various set relations such as (non)membership, subset or disjointness.
For example, ...

## Catalano-Fiore Vector Commitments

A vector commitment (VC) scheme allows a prover with access to a vector $\mathbf{v} = [ v_1, \dots, v_n ]$ to convince any verifier that position $i$ in $\mathbf{v}$ stores $v_i$ for any index $i\in[n]$.
Importantly, verifers only store a succinct digest of the vector (e.g., a 32-byte hash) rather than the full vector $\mathbf{v}$.

## Linear Diophantine Equations

Equations of the form $\sum_i a_i x_i = 0$ where the $x_i$’s are integer unknowns are called linear Diophantine equations.
Their integer solutions can be computed using greatest common denominator (GCD) tricks.
In this post, we go over a few basic types of such equations and their integer solutions.

22 post articles, 3 pages.