Quantum Information & Computation

The theory of divisors has played a remarkably productive role in classical cryptography, particularly in the application of function fields from algebraic geometry. Asymmetric (public) key systems based upon the Discrete Logarithm Problem [DLP] for finite fields, the Elliptic Curve DLP and more recently the Hyperelliptic Curve DLP have each depended heavily upon mathematical structures reliant upon the theory of divisors as indeed have systems based upon the Integer Factorisation Problem.

With the advent of Shors algorithms, and the Hidden Subgroup Problem the continuing successful application of finite fields in asymmetric key cryptography has been put into question. Recent workshops within the classical community however, have cited various schemes that may be resistant to such quantum attacks (for example the Diffie-Lamport-Merkle Signature system, the NTRU encryption system, the McEliece encryption system, and the HFE signature system). Additionally, applications of Galois theory to classical protocols are not restricted solely to their use in asymmetric key cryptography but may be found for example, in error detection and symmetric (private) key cryptography. In quantum applications such as coding, cryptography, tomography and computing an increasing role is being played by concepts drawn from algebraic geometry.

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