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Title: Post quantum cryptography : alternative solutions to hard problems for security
Authors: Emerole, Kelechi Chukwunonyerem
Issue Date: 2023
Publisher: Newcastle University
Description: Post Quantum cryptography are defined as public key crypto algorithms whose pub lic keys are generated from hard computational problems that are complex to solve in polynomial time by a quantum computer given worst case instances. The hard problems which have been proven to be quantum resistant include the shortest vec tor problem of lattices, the syndrome decoding problem of certain error correcting codes and the isomorphism of polynomial problem of multivariate quadratic poly nomials. Solutions to these problems have been proposed which in turn have impact on the security and storage cost of such algorithms to protect information systems in the future. In this thesis, alternative solutions are proposed which are based on robust and complex vector space mappings. Firstly, Dimensionality mapping is pro posed to reduce the basis into its linear independent vectors at low dimensionality by constructing a collapse function as an optimization problem. This optimization problem can be solved on the condition that a projection of the basis vectors from the High dimensional space to low dimensional manifold would have nearly orthogonal constitution. These eliminates the need for pre-processing using Gram-Schmidt Or thogonalization process. Implementing this approach on a channel basis, showed an improved BER performance over the Lenstra-Lenstra-Lovatsz algorithm for about 1db and 4db in the 4 × 4 and 6 × 6 uncoded system using 4QAM constellation. Secondly, the solution of the syndrome decoding problem is generalized to codes associated with the totally non-negative Grassmannian. The solution was reduced to an instance of finding a subset of the Pl¨ucker coordinates with the minimum Grassmann distance from the subspace containing the encrypted message symbols. Furthermore, bounds where derived which showed that the complexity scales up on the size of the Pl¨ucker coordinates. In addition, experimental results on decoding failure probability and complexity based on row operations where presented and compared to Low Density parity check codes in the Hamming metric. Finally, the kernel function of the New Mersenne number transform was applied to hide the 4 structure of the core map(central polynomial) of a multivariate polynomial based cryptosystem. This is in order to mitigate the interpolation of the rank of the quadratic form by an adversary. The implementation of this new isomorphism from the New Mersenne Number Transform showed an average of 69% reduction in secret key size. Further implementation of the isomorphism against key recovery attacks from the MinRank instance where carried out and it was shown that for lower field sizes the new isomorphism had an average success time of 13.8%.
Appears in Collections:School of Electrical, Electronic and Computer Engineering

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