Handling Vinegar Variables to Shorten Rainbow Key Pairs

Multivariate quadratic equations are the basis of one of the main mathematical techniques for the creation of digital signatures that are quantum-resistant. In these schemes, the creation and verification of signatures is highly efficient. However, key sizes are quite impractical and orders of magnitude greater than conventional schemes. One of the best-known signature schemes built upon multivariate equations is called Rainbow, which is based on the Oil-Vinegar principle. We observe that the reuse of vinegar variables in the signature generation step of the Rainbow scheme leads to a shorter representation of its central map, and thus, of the entire private key. We analyse the security implications of this strategy and present a modification to the Rainbow scheme, enabling a private key size reduction of up to \(85\%\) with secure parameters. Additionally, this framework can be applied on top of already existing schemes that shorten either private or public keys, spawning derivatives that reduce the total key pair size by a factor of 3.5.