Subfield attack: leveraging composite-degree extensions in the Quotient Ring transform

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The VOX signature scheme is a multivariate signature scheme which was submitted to the Round 1 Additional Digital Signature Schemes NIST process. VOX relies on the Hat Plus perturbation and the Quotient-Ring transform (QR). We formalize a dimension criterion enabling the direct attack to be used as a key recovery attack against UOV schemes. This enables a practical cryptanalysis of the Round 1 VOX parameters. Next, we show that some of the alternative parameters proposed for VOX after attack