Projective interpolation of polynomial vectors and improved key recovery attack on SFLASH

SFLASH is an instance of the famous C* \(^{-}\) multivariate public key cryptographic schemes and it was chosen by the NESSIE cryptographic project of the European Consortium in 2003 as a candidate signature algorithm used for digital signatures on limited-resource devices. Recently, a successful private key recovery attack on SFLASH was proposed by Bouillaguet, Fouque and Macario-Rat by uncovering the kernel properties of quadratic forms of the central map. The most expensive step in the attack is the calculation of kernel vectors of skew-symmetric matrices over a bivariate polynomial ring. Bouillaguet et al. proposed two methods to accomplish this computation. Both methods involve symbolic computation on bivariate polynomials. The first method computes characteristic polynomials of matrices of polynomials and is very expensive. The second method involves a Gröbner basis computation and so its complexity is difficult to estimate. In this paper, we show this critical step of calculating kernel vectors can be done by numerical computation on field elements instead of symbolic computation. Our method uses a nondeterministic interpolation of polynomial vectors called projective interpolation, and its complexity can be explicitly evaluated. Experiments show that it is much faster, making the total attack on SFLASH about 30 times faster (the critical step is about 100 times faster) than the first method of Bouillaguet et al. The new method is also slighter faster than their second method.