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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...

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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...

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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...

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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, ...

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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.

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