Superfluous Keys in Multivariate Quadratic Asymmetric Systems

In this article, we show that public key schemes based on multivariate quadratic equations allow many equivalent, and hence super∞uous private keys. We achieve this result by investigating several transformations to identify these keys and show their application to Hidden Field Equations (HFE), C ⁄ , and Unbalanced Oil and Vinegar schemes (UOV). In all cases, we are able to reduce the size of the private | and hence the public | key space by at least one order of magnitude. We see applications of our technique both in cryptanalysis of these schemes and in memory e‐cient implementations.

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