Roots of Square: Cryptanalysis of Double-Layer Square and Square+

Square is a multivariate quadratic encryption scheme proposed in 2009. It is a specialization of Hidden Field Equations by using only odd characteristic elds and also X as its central map. In addition, it uses embedding to reduce the number of variables in the public key. However, the system was broken at Asiacrypt 2009 using a di erential attack. At PQCrypto 2010 Clough and Ding proposed two new variants named Double-Layer Square and Square+. We show how to break Double-Layer Square using a re ned MinRank attack in 2 eld operations. A similar fate awaits Square+ as it will be broken in 2 eld operations using a mixed MinRank attack over both the extension and the ground eld. Both attacks recover the private key, given access to the public key. We also outline how possible variants such as Square or multi-Square can be attacked.