Several classes of 0-APN power functions over F2n

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arXiv:2210.15103v1 [cs.IT] 27 Oct 2022

Tao Fua, Haode Yana,∗

aSchool of Mathematics, Southwest Jiaotong University, Chengdu, 610031, China

Abstract

Recently, the investigation of Partially APN functions has attracted a lot of attention. In this

paper, with the help of resultant elimination and MAGMA, we propose several new inﬁnite

classes of 0-APN power functions over F2n. By the main result in [4], these 0-APN power

functions are CCZ-inequivalent to the known ones. Moreover, these inﬁnite classes of 0-APN

power functions can explain some exponents for 1 ≤n≤11 which are not yet “explained” in the

tables of Budaghyan et al. [3].

Keywords: APN function, 0-APN function, Power function, Resultant

2000 MSC: 11T06, 94A60

1. Introduction

Let F2nbe the ﬁnite ﬁeld with 2nelements and F∗

2n=F2n\ {0}, where nis a positive integer.

Let Fbe a function from F2nto itself. The derivative function of Fwith respect to any a∈F2n

is the function DaFfrom F2nto F2ngiven by DaF(x) = F(x+a) + F(x), where x∈F2n. For

any a∈F∗

2nand b∈F2n, let NF(a,b)denote the number of solutions x∈F2nof DaF(x) = b.

The differential uniformity of Fis deﬁne as ∆F=maxa∈F∗

2n,b∈F2nNF(a,b). We call the function

Fdifferentially k-uniform if ∆F=k. We expect ∆Fto achieve a smaller value when Fis used

for resisting differential attacks. Note that the solutions of DaF(x) = bcome in pairs, then the

smallest ∆Fis 2. When ∆F=2, Fis called an almost perfect nonlinear (APN for short) function.

Currently known only 6 inﬁnite classes of APN power functions over F2nare given in [5]-[9],

[11].

In order to study the conjecture of the highest possible algebric degree of APN functions, the

concept of partially APN functions was ﬁrst introduced by Budaghyan et al. [2] as follows.

Deﬁnition 1.1. ([2]) Let F be a function from F2nto itself. For a ﬁxed x0∈F2n, we call F is x0-

APN (or partially APN) if all the points u,v satisfying F (x0) + F(u) + F(v) + F(x0+u+v) = 0

belong to the curve (x0+u)(x0+v)(u+v) = 0.

∗Corresponding author

Email addresses:

futao0923@163.com

(Tao Fu),

hdyan@swjtu.edu.cn

(Haode Yan)

Table 1: 0-APN power mappings F(x) = xdover F2nfor 1 ≤n≤11 which are not yet “explained”

n d

9 45,125

10 51,93, 105, 351, 447

11 59,93,169,243,303,507, 245,447, 89, 445

If Fis APN, it is obvious that Fis x0-APN for any x0∈F2n. Conversely, there are many

examples that are x0-APN for some x0∈F2nbut not APN. In[2], Budaghyan et al. provided

some propositions and characterizations of partial APN functions. Moreover, Pott proved that

for any n≥3, there are partial 0-APN permutations on F2nin [12]. When Fis a power mapping,

i.e., F(x) = xdfor some integer d, due to the nice algebric structure of F, we only need to

consider the partial APN properties of Fat x0=0 and x0=1. Moreover, Fis 0-APN if and

only if the equation F(x+1) + F(x) + 1=0 has no solution in F2n\F2. Many classes of 0-APN

but not APN power functions over F2nare constructed in [2] and [3]. They also list all power

functions F(x) = xdover F2nfor 1 ≤n≤11 that are 0-APN but not APN [3]. Recently, Qu

and Li provided seven classes of 0-APN power functions over F2n, some of them were proved to

be locally-APN [13]. Very recently, Wang and Zha proposed several new inﬁnite classes of 0-

APN power function by using the multivariate method and resultant elimination [14]. However,

some pairs of (d,n)in [3] are not yet “explained”, we summarize them in Table 1. In this paper,

we extend them to new inﬁnite families of 0-APN power functions. This paper is organized

as follows. Some basic results of the resultant of polynomials are introduced in Section 2. In

Section 3, we propose seven new inﬁnite classes of 0-APN power functions over F2n. We mention

that we follow the approach proposed in [13] and [14]. For the convenience, we summarize the

seven classes of 0-APN power functions in Table 2. By the main result in [4], the 0-APN power

functions obtained in this paper are CCZ-inequivalent to the known ones.

2. On the resultant of polynomials

In this section, in order to prove our results, we need concept and relevant conclusions of the

resultant of two polynomials.

Deﬁnition 2.1. ([10]) Let Kbe a ﬁeld, f (x) = a0xn+a1xn−1+···+an∈K[x]and g(x) = b0xm+

b1xm−1+···+am∈K[x]be two polynomials of degree n and m respectively, where n,m∈N.

Table 2: New classes of 0-APN power functions f(x) = xdover F2n

xdconditions (d,n)can be explained in Table 1Reference

x3·2k−7n=2k+1(89,11)Thm 3.1

x22k+1−2k+1−2k+1n=3k+1(105,10)Thm 3.2

x3(2k−1)n=2k,3∤k(93,10)Thm 3.3

x5(2k+1+2k+1)n=2k+1,k6≡ 2(mod 5) (125,9),(245,11)Thm 3.4

x3(2k−1)n=2k+1,k6≡ 13 (mod 27) (45,9),(93,11)Thm 3.5

x3(2k+1+1)n=3k+1,k6≡ 9(mod 14) (51,10)Thm 3.6

x−99∤n(447,10)Thm 3.7

Then the resultant Res(f,g)of the two polynomials is deﬁned by the determinant

Res(f,g) =



a0a1··· an0··· 0

0a0a1··· an0··· 0

0··· 0a0a1··· an

b0b1··· bm0··· 0

0b0b1··· bm··· 0

0··· 0b0b1··· bm













m rows











n rows

of order m +n.

If the degree of fis nand f(x) = a0(x−α1)(x−α2)···(x−αn)in the splitting ﬁeld of f

over K, then Res(f,g)is also given by the formula

Res(f,g) = am

∏

i=1

g(αi).

In this case, we have Res(f,g) = 0 if and only if fand ghave a common root, which means that

fand ghave a common divisor in K[x]of positive degree.

For two polynomials F(x,y),G(x,y)∈K[x,y]of positive degree in y, the resultant Res(F,G,y)

of Fand Gwith respect to yis the resultant of Fand Gwhen considered as polynomials in the

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摘要：

arXiv:2210.15103v1[cs.IT]27Oct2022Severalclassesof0-APNpowerfunctionsoverF2nTaoFua,HaodeYana,∗aSchoolofMathematics,SouthwestJiaotongUniversity,Chengdu,610031,ChinaAbstractRecently,theinvestigationofPartiallyAPNfunctionshasattractedalotofattention.Inthispaper,withthehelpofresultanteliminationandMAGMA...

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Several classes of 0-APN power functions over F2n

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