Key-Recovery Attack on the ASASA Cryptosystem with Expanding S-Boxes

We present a cryptanalysis of the ASASA public key cipher introduced at Asiacrypt 2014 [3]. This scheme alternates three layers of affine transformations A with two layers of quadratic substitutions S. We show that the partial derivatives of the public key polynomials contain information about the intermediate layer. This enables us to present a very simple distinguisher between an ASASA public key and random polynomials. We then expand upon the ideas of the distinguisher to achieve a full secret key recovery. This method uses only linear algebra and has a complexity dominated by the cost of computing the kernels of 2(26) small matrices with entries in F-16.