The ASASA construction is a new design scheme introduced at Asiacrypt 2014 by Biryukov, Bouillaguet and Khovratovich. Its versatility was illustrated by building two public-key encryption schemes, a secret-key scheme, as well as super S-box subcomponents of a white-box scheme. However one of the two public-key cryptosystems was recently broken at Crypto 2015 by Gilbert, Plût and Treger. As our main contribution, we propose a new algebraic key-recovery attack able to break at once the secret-key scheme as well as the remaining public-key scheme, in time complexity 2^{63} and 2^{39} respectively (the security parameter is 128 bits in both cases). Furthermore, we present a second attack of independent interest on the same public-key scheme, which heuristically reduces the problem of breaking the scheme to an LPN instance with tractable parameters. This allows key recovery in time complexity 2^{56}. Finally, as a side result, we outline a very efficient heuristic attack on the white-box scheme, which breaks instances claiming 64 bits of security under one minute on a laptop computer.