Cryptanalysis of the quaternion rainbow

Abstract

Rainbow is one of the signature schemes based on multivariate problems. While its signature generation and verification are fast and the security is presently sufficient under suitable parameter selections, the key size is relatively large. Recently, Quaternion Rainbow - Rainbow over quaternion ring - was proposed by Yasuda, Sakurai and Takagi (CT-RSA'12) to reduce the key size of Rainbow without impairing the security. However, a new vulnerability emerges from the structure of quaternion ring; in fact, Thomae (SCN'12) found that Quaternion Rainbow is less secure than the same-size original Rainbow. In the present paper, we further study the security of Quaternion Rainbow and get better security results than Thomae's ones. Especially, we find that Quaternion Rainbow over even characteristic field, whose security level is estimated as about the original Rainbow of at most 3/4 by Thomae's analysis, is almost as secure as the original Rainbow of at most 1/4-size. © 2013 Springer-Verlag.