A Matrix Approach for Constructing Quadratic APN Functions

Yuyin Yu; Mingsheng Wang; Yongqiang Li

Paper 2013/007

A Matrix Approach for Constructing Quadratic APN Functions

Yuyin Yu, Mingsheng Wang, and Yongqiang Li

Abstract

We find a one to one correspondence between quadratic APN functions without linear or constant terms and a special kind of matrices (We call such matrices as QAMs). Based on the nice mathematical structures of the QAMs, we have developed efficient algorithms to construct quadratic APN functions. On $\mathbb{F}_{2^7}$, we have found more than 470 classes of new CCZ-inequivalent quadratic APN functions, which is 20 times more than the known ones. Before this paper, there are only 23 classes of CCZ-inequivalent APN functions on $\mathbb{F}_{2^{8}}$ have been found. With our method, we have found more than 2000 classes of new CCZ-inequivalent quadratic APN functions, and this number is still increasing quickly.

Note: We give more APN Functions on GF(256) in Appendix 2.

Metadata

Available format(s): PDF
Publication info: Published elsewhere. Unknown status
Keywords: APN quadratic functions EA-equivalence CCZ-equivalence.
Contact author(s): yuyuyin @ 163 com
History: 2015-04-27: last of 2 revisions; 2013-01-11: received; See all versions
Short URL: https://ia.cr/2013/007
License: CC BY

BibTeX

@misc{cryptoeprint:2013/007,
      author = {Yuyin Yu and Mingsheng Wang and Yongqiang Li},
      title = {A Matrix Approach for Constructing Quadratic APN Functions},
      howpublished = {Cryptology ePrint Archive, Paper 2013/007},
      year = {2013},
      note = {\url{https://eprint.iacr.org/2013/007}},
      url = {https://eprint.iacr.org/2013/007}
}