On the Number of Distinct Values of a Class of Functions with Finite Domain

Coulter, Robert S.; Senger, Steven

doi:10.1007/s00026-014-0220-2

“…In previous sections we used the value N ( f ) to obtain a lower bound for the image size of some special maps f . In [21] an upper bound for | Im( f )| depending on N ( f ) is found. The upper bound from is valid for maps between arbitrary finite sets, however we state it here only for the binary finite fields.…”

Section: Upper Bounds On the Image Sets Of Apn Mapsmentioning

confidence: 97%

Image sets of perfectly nonlinear maps

Kölsch

¹

,

Kriepke

²

,

Kyureghyan

³

2022

Des. Codes Cryptogr.

2

0

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We consider image sets of differentially d-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution. Further, we focus on a particularly interesting case of APN maps on binary fields $$\mathbb {F}_{2^n}$$ F 2 n . We show that APN maps with the minimal image size are very close to being 3-to-1. We prove that for n even the image sets of several important families of APN maps are minimal, and as a consequence they have the classical Walsh spectrum. Finally, we present upper bounds on the image size of APN maps. For a non-bijective almost bent map f, these results imply $$\frac{2^n+1}{3}+1 \le |{\text {Im}}(f)| \le 2^n-2^{(n-1)/2}$$ 2 n + 1 3 + 1 ≤ | Im ( f ) | ≤ 2 n - 2 ( n - 1 ) / 2 .

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“…So far, we used the value N (f ) to obtain a lower bound for the image size of a special map. In [18], information on N (f ) was used to find an upper bound for the image size of planar maps. Using Proposition 6.7, we can do the same for almost bent maps.…”

Section: -Divisible Apn Mapsmentioning

confidence: 99%

Image sets of perfectly nonlinear maps

Kölsch¹,

Kriepke²,

Kyureghyan³

2020

Preprint

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We present a lower bound on the image size of a d-uniform map, d ≥ 1, of finite fields, by extending the methods used for planar maps in [22,26,17]. In the particularly interesting case of APN maps on binary fields, our bound coincides with the one obtained in [19], which is proved with a linear programming method.We study properties of APN maps of F 2 n with minimal image set. In particular, we observe that for even n, a Dembowski-Ostrom polynomial of form f (x) = f ′ (x 3 ) is APN if and only if f is almost-3-to-1, that is when its image set is minimal. We show that any almost-3-to-1 quadratic map is APN, if n is even. For n odd, we present APN Dembowski-Ostrom polynomials on F 2 n with image sizes 2 n−1 and 5 • 2 n−3 .We present several results connecting the image sets of special APN maps with their Walsh spectrum. Especially, we show that a large class of APN maps has the classical Walsh spectrum. Finally, we prove that the image size of a non-bijective almost bent map contains at most 2 n − 2 (n−1)/2 elements.

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“…7 (1,6), (2,21), (3,16), (4,15), (5,18), (6,11) 8 (1, 7), ({2, 4}, 28), ({3, 5}, 21), (6, 28), (7,13) 9 (1,8), (2,36), (3,24), (4, 30), (5,24), (6, 28) ⋆ , (7,32), (8,15) 11 ( are based on the fact that by choosing a basis of F p s over F p , a polynomial over F p s can be represented as a mapping from vector space F s p to F s p . In the definition of EA and CCZ equivalence, the structure of the vector space F s p plays a crucial role.…”

mentioning

confidence: 99%

“…Finally, we mention an interesting work due to Coulter and Senger [16]. Given two sets A and B, the authors considered a function f : A → B, and defined N 2 (f ) to be the number of pairs (x, y) ∈ A × A, such that x = y and f (x) = f (y).…”

mentioning

confidence: 99%

“…Given two sets A and B, the authors considered a function f : A → B, and defined N 2 (f ) to be the number of pairs (x, y) ∈ A × A, such that x = y and f (x) = f (y). Some bounds on the size of the value set {f (x) | x ∈ A}, which involves the number N 2 (f ), were derived in [16]. A very recent work due to Ding and Tang [17] employs polynomial f over finite field to construct combinatorial t-designs.…”

mentioning

confidence: 99%

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Intersection distribution, non-hitting index and Kakeya sets in affine planes

Li¹,

Pott²

2020

Preprint

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In this paper, we propose the concepts of intersection distribution and non-hitting index, which can be viewed from two related perspectives. The first one concerns a point set S of size q + 1 in the classical projective plane P G(2, q), where the intersection distribution of S indicates the intersection pattern between S and the lines in P G(2, q). The second one relates to a polynomial f over a finite field Fq, where the intersection distribution of f records an overall distribution property of a collection of polynomials {f (x) + cx | c ∈ Fq}. These two perspectives are closely related, in the sense that each polynomial produces a (q +1)-set in a canonical way and conversely, each (q +1)-set with certain property has a polynomial representation. Indeed, the intersection distribution provides a new angle to distinguish polynomials over finite fields, based on the geometric property of the corresponding (q + 1)-sets. Among the intersection distribution, we identify a particularly interesting quantity named non-hitting index.For a point set S, its non-hitting index counts the number of lines in P G(2, q) which do not hit S. For a polynomial f over a finite field Fq, its non-hitting index gives the summation of the sizes of q value sets {f (x) + cx | x ∈ Fq}, where c ∈ Fq. We derive lower and upper bounds on the non-hitting index and show that the non-hitting index contains much information about the corresponding set and the polynomial. More precisely, using a geometric approach, we show that the non-hitting index is sufficient to characterize the corresponding point set and the polynomial, when it is very close to the lower and upper bounds. Moreover, we employ an algebraic approach to derive the non-hitting index and the intersection distribution of several families of point sets and polynomials. As an application, we consider the determination of the sizes of Kakeya sets in affine planes. The polynomial viewpoint of intersection distributions enable us to compute the size of a few families of Kakeya sets with nice algebraic properties.Finally, we describe the connection between these new concepts and various known results developed in different contexts and propose some future research problems.

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On the Number of Distinct Values of a Class of Functions with Finite Domain

Cited by 12 publications

References 26 publications

Image sets of perfectly nonlinear maps

Image sets of perfectly nonlinear maps

Image sets of perfectly nonlinear maps

Intersection distribution, non-hitting index and Kakeya sets in affine planes

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