Abstract
It has been proved in Bierbrauer and Kyureghyan (Des. Codes Cryptogr. 46:269–301, 2008) that a binomial function aX i + bX j can be crooked only if both exponents i, j have 2-weight ≤2. In the present paper we give a brief construction for all known examples of crooked binomial functions. These consist of an infinite family and one sporadic example. The construction of the sporadic example uses the properties of an algebraic curve of genus 3. Computer experiments support the conjecture that each crooked binomial is equivalent either to a member of the family or to the sporadic example.
Similar content being viewed by others
References
Bierbrauer D.: Codes auf hyperelliptischen und trigonalen Kurven. University of Heidelberg, Diplomarbeit (2006)
Bierbrauer J.: Introduction to Coding Theory. Chapman and Hall, CRC Press (2004).
Bierbrauer J., Kyureghyan G.M.: Crooked binomials. Des. Codes Cryptogr 46, 269–301 (2008)
Budaghyan L., Carlet C., Felke P., Leander G.: An infinite class of quadratic APN functions which are not equivalent to power mappings. In: Proceedings of the IEEE International Symposium on Information Theory, Seattle (2006).
Budaghyan L., Carlet C., Leander G.: A class of quadratic APN binomials inequivalent to power functions (submitted).
Budaghyan L., Carlet C., Leander G.: Another class of quadratic APN binomials over \({\mathbb{F}_{2^n}:}\) the case n divisible by 4 (manuscript).
Edel Y., Kyureghyan G., Pott A.: A new APN function which is not equivalent to a power mapping. IEEE Trans. Inform. Theory 52, 744–747 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Pott.
Rights and permissions
About this article
Cite this article
Bierbrauer, J. A family of crooked functions. Des. Codes Cryptogr. 50, 235–241 (2009). https://doi.org/10.1007/s10623-008-9227-1
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-008-9227-1