New pseudo-planar binomials in characteristic two and related schemes

Hu, Sihuang; Li, Shuxing; Zhang, Tao; Feng, Tao; Ge, Gennian

doi:10.1007/s10623-014-9958-0

Cited by 10 publications

(11 citation statements)

References 23 publications

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“…Therefore these functions should be equivalent to F (x) = 0. The same argument also works for the functions in Result 3 discovered by Hu et al [13].…”

Section: )supporting

confidence: 66%

“…We will introduce a technique. It is generalized from a trick which was firstly used in the proof of [8, Theorem 3.1] and then in the proof of [13,Proposition 3.6]. Let us set up the following notations.…”

Section: B Discussing Det M Bmentioning

confidence: 99%

“…In [13,Proposition 3.2], a sufficient and necessary condition for F to be pseudo-planar was given as follows.…”

Section: Furthermentioning

confidence: 99%

“…Then we use this new approach to construct new explicit families of quadratic pseudo-planar functions over F 2 n , and reconstruct known families. The constructions are split into three cases according to the values of the extension degree t. For the case of extension degree t = 3, we construct three families of pseudo-planar functions, and study a family of trinomial, which is a generalization of the three families of functions in [13]. The monomial polynomial is also revisited, and a sufficient and necessary condition for it to be pseudo-planar is given.…”

Section: Introductionmentioning

confidence: 99%

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A New Approach to Constructing Quadratic Pseudo-Planar Functions Over $ {\mathbb F}_{2^{n}}$

2016

IEEE Trans. Inform. Theory

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Planar functions over finite fields give rise to finite projective planes. They were also used in the constructions of DES-like iterated ciphers, error-correcting codes, and codebooks. They were originally defined only in finite fields with odd characteristic, but recently Zhou introduced pesudo-planar functions in even characteristic which yields similar applications. All known pesudo-planar functions are quadratic and hence they give presemifields. In this paper, a new approach to constructing quadratic pseudo-planar functions is given. Then five explicit families of pseudo-planar functions are constructed, one of which is a binomial, two of which are trinomials, and the other two are quadrinomials. All known pesudo-planar functions are revisited, some of which are generalized. These functions not only lead to projective planes, relative difference sets and presemifields, but also give optimal codebooks meeting the Levenstein bound, complete sets of mutually unbiased bases (MUB) and compressed sensing matrices with low coherence.

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“…Therefore these functions should be equivalent to F (x) = 0. The same argument also works for the functions in Result 3 discovered by Hu et al [13].…”

Section: )supporting

confidence: 66%

Section: B Discussing Det M Bmentioning

confidence: 99%

“…In [13,Proposition 3.2], a sufficient and necessary condition for F to be pseudo-planar was given as follows.…”

Section: Furthermentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A New Approach to Constructing Quadratic Pseudo-Planar Functions Over $ {\mathbb F}_{2^{n}}$

2016

IEEE Trans. Inform. Theory

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“…Most notably, Müller and Zieve [21] give a characterisation of low-degree planar monomials, thereby proving Conjecture 3.4 and providing a different proof of Theorem 1.3. New examples of planar functions, in particular planar binomials, have been found by Hu, Li, Zhang, Feng, and Ge in [13].…”

Section: Final Remarksmentioning

confidence: 93%

Planar functions over fields of characteristic two

Schmidt

Zhou

2014

J Algebr Comb

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Abstract. Classical planar functions are functions from a finite field to itself and give rise to finite projective planes. They exist however only for fields of odd characteristic. We study their natural counterparts in characteristic two, which we also call planar functions. They again give rise to finite projective planes, as recently shown by the second author. We give a characterisation of planar functions in characteristic two in terms of codes over Z4. We then specialise to planar monomial functions f (x) = cx t and present constructions and partial results towards their classification. In particular, we show that t = 1 is the only odd exponent for which f (x) = cx t is planar (for some nonzero c) over infinitely many fields. The proof techniques involve methods from algebraic geometry.

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Investigating rational perfect nonlinear functions

Bartoli

Timpanella

2023

Annali di Matematica

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Perfect nonlinear (PN) functions over a finite field, whose study is also motivated by practical applications to Cryptography, have been the subject of several recent papers where the main problems, such as effective constructions and non-existence results, are considered. So far, all contributions have focused on PN functions represented by polynomials, and their constructions. Unfortunately, for polynomial PN functions, the approach based on Hasse–Weil type bounds applied to function fields can only provide non-existence results in a small degree regime. In this paper, we investigate the non-existence problem of rational perfect nonlinear functions over a finite field. Our approach makes it possible to use deep results about the number of points of algebraic varieties over finite fields. Our main result is that no PN rational function f/g with $$f,g\in \mathbb {F}_q[X]$$ f , g ∈ F q [ X ] exists when certain mild arithmetical conditions involving the degree of f(X) and g(X) are satisfied.

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New pseudo-planar binomials in characteristic two and related schemes

Cited by 10 publications

References 23 publications

A New Approach to Constructing Quadratic Pseudo-Planar Functions Over $ {\mathbb F}_{2^{n}}$

A New Approach to Constructing Quadratic Pseudo-Planar Functions Over $ {\mathbb F}_{2^{n}}$

Planar functions over fields of characteristic two

Investigating rational perfect nonlinear functions

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