MIRANDA: short signatures from a leakage-free full-domain-hash scheme

Abstract

We present $\mathsf{Miranda}$, the first family of full-domain-hash signatures based on matrix codes. This signature scheme fulfils the paradigm of Gentry, Peikert and Vaikuntanathan ($\mathsf{GPV}$), which gives strong security guarantees. Our trapdoor is very simple and generic: if we propose it with matrix codes, it can actually be instantiated in many other ways since it only involves a subcode of a decodable code (or lattice) in a unique decoding regime of parameters. Though $\mathsf{Miranda}$ signing algorithm relies on a decoding task where there is exactly one solution, there are many possible signatures given a message to sign and we ensure that signatures are not leaking information on their underlying trapdoor by means of a very simple procedure involving the drawing of a small number of uniform bits. In particular $\mathsf{Miranda}$ does not use a rejection sampling procedure which makes its implementation a very simple task contrary to other $\mathsf{GPV}$-like signatures schemes such as $\mathsf{Falcon}$ or even $\mathsf{Wave}$. We instantiate $\mathsf{Miranda}$ with the famous family of Gabidulin codes represented as spaces of matrices and we study thoroughly its security (in the EUF-CMA security model). For~$128$ bits of classical security, the signature sizes are as low as~$90$ bytes and the public key sizes are in the order of~$2.6$ megabytes.