Multivariate Cryptography

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Synonyms

MPKC; \(\mathcal{M}\mathcal{Q}\); Multivariate quadratic public-key Cryptosystem (MQPKC)

Related Concepts

Differential Cryptanalysis; Digital Signature Schemes; Post-quantum Cryptography; Public-key Cryptography

Definition

A Multivariate Public-Key Cryptosystem (MPKC) is a public-key cryptosystem where the public map \(\mathcal{P}\), or trapdoor one-way function, is given as a set of m polynomial equations of a small degree d over n variables in a finite fieldF. Usually d = 2, hence the alternate name “Multivariate Quadratic” (MQ).

To decrypt, authenticate, or sign digitally, a user must, for a given m-tuple \(\mathbf{z} = ({z}_{1},\ldots, {z}_{m})\), find a solution \(\mathbf{w} = ({w}_{1},\ldots, {w}_{n})\) of the system $$(\mathcal{P})\left \{\begin{array}{l} {p}_{1}({w}_{1},\ldots, {w}_{n}) = {z}_{1} \\ \cdots \\ {p}_{m}({w}_{1},\ldots, {w}_{n}) = {z}_{m} \end{array} \right..$$

For a digital signature, a challenge–response authentication scheme,