At SAC 2009, Misoczki and Barreto proposed a new class of codes, which have parity-check matrices that are quasi-dyadic. A special subclass of these codes were shown to coincide with Goppa codes and those were recommended for cryptosystems based on error-correcting codes. Quasi-dyadic codes have both very compact representations and allow for efficient processing, resulting in fast cryptosystems with small key sizes. In this paper, we generalize these results and introduce quasimonoidic codes, which retain all desirable properties of quasi-dyadic codes. We show that, as before, a subclass of our codes contains only Goppa codes or, for a slightly bigger subclass, only Generalized Srivastava codes. Unlike before, we also capture codes over fields of odd characteristic. These include wild Goppa codes that were proposed at SAC 2010 by Bernstein, Lange, and Peters for their exceptional error-correction capabilities. We show how to instantiate standard code-based encryption and signature schemes with our codes and give some preliminary parameters.
In this article, we propose a new lattice-based threshold ring signature scheme, modifying Aguilar's code-based solution to use the short integer solution (SIS) problem as security assumption, instead of the syndrome decoding (SD) problem. By applying the CLRS identification scheme, we are also able to have a performance gain as result of the reduction in the soundness error to 1/2 per round. Such gain is also maintained through the application of the Fiat-Shamir heuristics to derive signatures from our identification scheme. From security perspective we also have improvements, because our scheme exhibits a worst-case to average-case reduction typical of lattice-based cryptosystems. This gives us confidence that a random choice of parameters results in a system that is hard to break, in average.
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