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

Enrico Thomae; Christopher Wolf

Paper 2011/431

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

Enrico Thomae and Christopher Wolf

Abstract

Square is a multivariate quadratic encryption scheme proposed in 2009. It is a specialization of Hidden Field Equations by using only odd characteristic fields and also X^2 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 differential 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 refined MinRank attack in 2^45 field operations. A similar fate awaits Square+ as it will be broken in 2^32 field operations using a mixed MinRank attack over both the extension and the ground field. 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.

Metadata

Available format(s): PDF
Publication info: Published elsewhere. PQCrypto 2011
Keywords: Multivariate Cryptography Algebraic Cryptanalysis Square Double-Layer Square Square+MinRank Key Recovery
Contact author(s): enrico thomae @ rub de
christopher wolf @ rub de
History: 2011-10-04: revised; 2011-08-12: received; See all versions
Short URL: https://ia.cr/2011/431
License: CC BY

BibTeX

@misc{cryptoeprint:2011/431,
      author = {Enrico Thomae and Christopher Wolf},
      title = {Roots of Square: Cryptanalysis of Double-Layer Square and Square+},
      howpublished = {Cryptology {ePrint} Archive, Paper 2011/431},
      year = {2011},
      url = {https://eprint.iacr.org/2011/431}
}