More infinite classes of APN-like Power Functions

@article{Qu2022MoreIC,
  title={More infinite classes of APN-like Power Functions},
  author={Longjiang Qu and Kangquan Li},
  journal={ArXiv},
  year={2022},
  volume={abs/2209.13456}
}
In the literature, there are many APN-like functions that generalize the APN properties or are similar to APN functions, e.g. locally-APN functions, 0-APN functions or those with boomerang uniformity 2. In this paper, we study the problem of constructing infinite classes of APN-like but not APN power functions. For one thing, we find two infinite classes of locally-APN but not APN power functions over F 2 2 m with m even, i.e., F 1 ( x ) = x j (2 m − 1) with gcd( j, 2 m + 1) = 1 and F 2 ( x ) = x… 

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References

SHOWING 1-10 OF 22 REFERENCES

On the Niho type locally-APN power functions and their boomerang spectrum

: In this article, we focus on the concept of locally-APN-ness (“APN” is the abbrevia-tion of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which

The differential spectrum and boomerang spectrum of a class of locally-APN functions

The boomerang spectrum of the power mapping F ( x ) = x k ( q − 1) over F q 2, where q = p m, p is a prime, m is a positive integer and gcd( k, q +1) = 1, is determined.

Partially APN functions with APN-like polynomial representations

Several families of monomial functions with APN-like exponents that are not APN, but are partially 0-APN for infinitely many extensions of the binary field $$\mathbb {F}_2$$ F 2 .

Almost Perfect Nonlinear Power Functions on GF(2n): The Welch Case

The first case supports a well-known conjecture of Welch stating that for odd n=2m+1, the power function x/sup 2m+3/ is even maximally nonlinear or, in other terms, that the crosscorrelation function between a binary maximum-length linear shift register sequence and a decimation of that sequence by 2/sup m/+3 takes on precisely the three values -1, -1/spl plusmn/2/Sup m+1/.

Partially APN Boolean functions and classes of functions that are not APN infinitely often

A notion of partial APNness is defined and various characterizations and constructions of classes of functions satisfying this condition are found, connecting this notion to the known conjecture that APN functions modified at a point cannot remain APN.

Almost Perfect Nonlinear Power Functions on GF(2n): The Niho Case

Almost perfect nonlinear (APN) mappings are of interest for applications in cryptography We prove for odd n and the exponent d=22r+2r?1, where 4r+1?0modn, that the power functions xd on GF(2n) is

Finite Fields

  • K. Conrad
  • Mathematics
    Series and Products in the Development of Mathematics
  • 2004
This handout discusses finite fields: how to construct them, properties of elements in a finite field, and relations between different finite fields. We write Z/(p) and Fp interchangeably for the

On the Boomerang Uniformity of Cryptographic Sboxes

A more in-depth analysis of boomerang connectivity tables, by studying more closely differentially 4-uniform Sboxes and answering the above open question.

Boomerang Connectivity Table: A New Cryptanalysis Tool

A new tool is proposed called Boomerang Connectivity Table (BCT), which evaluates r in a systematic and easy-to-understand way when \(E_m\) is composed of a single S-box layer, and can detect a new switching effect.

New Results About the Boomerang Uniformity of Permutation Polynomials

This paper presents an equivalent technique to compute BCT and the boomerang uniformity, which seems to be much simpler than the original definition by Cid, and obtains another class of 4-uniform BCT permutation polynomials overTeX.