Niho Type Research Articles

On the Niho Type Locally-APN Power Functions and Their Boomerang Spectrum

This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> such that gcd(2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>m</i></sup> +1, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i></sup> +1) = 1, we investigate the locally-APN-ness property of the Niho type power function <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ) = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>s</i>(2^<i>m</i>-1)+1</sup> over the finite field F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2^2<i>m</i></sub> for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">s</i> = (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i></sup> + 1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> , where (2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>k</i></sup> + 1) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> denotes the multiplicative inverse modulo 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>m</i></sup> +1. By employing finer studies of the number of solutions of certain equations over finite fields, we prove that <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ) is locally-APN and determine its differential spectrum. We emphasize that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for 2 ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</i> ≤ 10. In addition, we also determine the boomerang spectrum of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F(x)</i> by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang and Li.

New permutation trinomials from Niho exponents over finite fields with even characteristic

In this paper, a class of permutation trinomials of Niho type over finite fields with even characteristic is further investigated. New permutation trinomials from Niho exponents are obtained from linear fractional polynomials over finite fields, and it is shown that the presented results are the generalizations of some earlier works.

Several classes of negabent functions over finite fields

Negabent functions as a class of generalized bent functions have attracted a lot of attention recently due to their applications in cryptography and coding theory. In this paper, we consider the constructions of negabent functions over finite fields. First, by using the compositional inverses of certain binomial and trinomial permutations, we present several classes of negabent functions of the form $f(x)=\Tr_1^n(\lambda x^{2^k+1})+\Tr_1^n(ux)\Tr_1^n(vx)$, where $\lambda\in \F_{2^n}$, $2\leq k\leq n-1$, $(u,v)\in \F^*_{2^n}\times \F^*_{2^n}$, and $\Tr_1^n(\cdot)$ is the trace function from $\F_{2^n}$ to $\F_{2}$. Second, by using Kloosterman sum, we prove that the condition for the cubic monomials given by Zhou and Qu (Cryptogr. Commun., to appear, DOI 10.1007/s12095-015-0167-0.) to be negabent is also necessary. In addition, a conjecture on negabent monomials whose exponents are of Niho type is given.

Some classes of complete permutation polynomials over $\mathbb{F}_q $

By using a powerful criterion for permutation polynomials, we give several classes of complete permutation polynomials over finite fields. First, two classes of complete permutation monomials whose exponents are of Niho type are presented. Second, for any odd prime p, we give a sufficient and necessary condition for a−1xd to be a complete permutation polynomial over \(\mathbb{F}_{p^{4k} } \), where \(d = \frac{{p^{4k} - 1}} {{p^k - 1}} + 1\) and \(a \in \mathbb{F}_{p^{4k} }^* \). Finally, we present a class of complete permutation multinomials, which is a generalization of recent work.

Weight distribution of cyclic codes with arbitrary number of generalized Niho type zeroes

Cyclic codes are an important class of linear codes, whose weight distribution have been extensively studied. Most previous results obtained so far were for cyclic codes with no more than three zeroes. Inspired by the works of Li et al. (Sci China Math 53:3279---3286, 2010; IEEE Trans Inf Theory 60:3903---3912, 2014), we study two families of cyclic codes over $${\mathbb F}_p$$Fp with arbitrary number of zeroes of generalized Niho type, more precisely $${\mathcal {C}_{(d_0,d_1,\ldots ,d_t)}^{(1)}}$$C(d0,d1,?,dt)(1) (for $$p=2$$p=2) of $$t+1$$t+1 zeroes, and $${\mathcal {C}_{(\widetilde{d}_1,\ldots ,\widetilde{d}_t)}^{(2)}}$$C(d~1,?,d~t)(2) (for any prime $$p$$p) of $$t$$t zeroes for any $$t$$t. We find that the first family has at most $$(2t+1)$$(2t+1) non-zero weights, and the second has at most $$2t$$2t non-zero weights. Their weight distribution are also determined in the paper.

THE NUMBER OF SOLUTIONS TO THE EQUATION (x + 1)d= xd+ 1

In this paper, we study the number of solutions to the equation <TEX>$(x+1)^d=x^d+1$</TEX>. This equation gives the value of the third power sum equation in case of Niho type exponents and is helpful in finding the distribution of the values <TEX>$C_d({\tau})$</TEX>. We provide the number of the solutions using the new method.

Further Results on Niho Bent Functions

This paper consists of two main contributions. First, the Niho bent function consisting of 2r exponents (discovered by Leander and Kholosha) is studied. The dual of the function is found and it is shown that this new bent function is not of the Niho type. Second, all known univariate representations of Niho bent functions are analyzed for their relation to the completed Maiorana-McFarland class M. In particular, it is proven that two families do not belong to the completed class M. The latter result gives a positive answer to an open problem whether the class H of bent functions introduced by Dillon in his thesis of 1974 differs from the completed class M.

New Sequences with Low Correlation and Large Family Size

In direct-sequence code-division multiple-access (DS-CDMA) communication systems and direct-sequence ultra wideband (DS-UWB) radios, sequences with low correlation and large family size are important for reducing multiple access interference (MAI) and accepting more active users, respectively. In this paper, a new collection of families of sequences of length pn-1, which includes three constructions, is proposed. The maximum number of cyclically distinct families without GMW sequences in each construction is π(pn-1)/n · π(pm-1)/ m, where p is a prime number, n is an even number, and n = 2m, and these sequences can be binary or polyphase depending upon choice of the parameter p. In Construction I, there are pn distinct sequences within each family and the new sequences have at most d + 2 nontrivial periodic correlation {-pm-1, -1, pm-1,2pm-1,···,dpm-1}. In Construction II, the new sequences have large family size p2n and possibly take the nontrivial correlation values in {-pm-1, -1, pm-1, 2pm-1,···,(3d-4)pm-1}. In Construction III, the new sequences possess the largest family size p(d-1)n and have at most 2d correlation levels {-pm-1, -1, pm-1,2pm-1, ···,(2d-2)pm-1}. Three constructions are near-optimal with respect to the Welch bound because the values of their Welch-Ratios are moderate, WR ≐ d, WR ≐ 3d-4 and WR ≐ 2d-2, respectively. Each family in Constructions I, II and III contains a GMW sequence. In addition, Helleseth sequences and Niho sequences are special cases in Constructions I and III, and their restriction conditions to the integers m and n, pm ≠ 2 (mod 3) and n ≡ 0 (mod 4), respectively, are removed in our sequences. Our sequences in Construction III include the sequences with Niho type decimation 3 · 2m-2, too. Finally, some open questions are pointed out and an example that illustrates the performance of these sequences is given.

On Four-Valued Niho-Type Cross-Correlation Functions of $m$-Sequences

Consider the cross-correlation function Cd(tau) between two m-sequences of period p2k-1 that differ by a decimation d. Assume that d is of Niho type and Cd(tau) is four-valued. In this correspondence, the values of Cd(tau) are described. If p=2 then the values of Cd(tau) are -1-2 k,-1,-1+2k, and -1+2k+j for some integer j>0 for which 2j divides k. If p>2 then the values are -1-pk,-1,-1+pk, and -1+2middotpk

Construction of generalized binary Bent sequences

Bent functions in trace forms play an important role in the constructions of generalized binary Bent sequences. Trace representation of some degree two Bent functions are presented in this paper. A sufficient and necessary condition is derived to determine whether the sum of the combinations of Gold functions, \(tr_1^n (x^{2^i + 1} )\), 1≤i≤n−1, over finite fields \(F_{2^n } \) (n be even) in addition to another term \(tr_1^{n/2} (x^{2n/2 + 1} )\) is a Bent function. Similar to the result presented by Khoo et al., the condition can be verified by polynominal greatest common divisor (GCD) computation. A similar result also holds in the case \(F_{p^n } \) (n be even, p be odd prime). Using the constructed Bent functions and Niho type Bent functions given by Dobbertin et al., many new generalized binary Bent sequences are obtained.

Niho type cross-correlation functions via dickson polynomials and Kloosterman sums

Suppose that n=2k is even. We study the cross-correlation function between two m-sequences for Niho type decimations d=(2/sup k/-1)s+1. We develop a new technique to study the value distribution of these cross-correlation functions, which makes use of Dickson polynomials. As a first application, we derive here the distribution of the six-valued cross-correlation function for s=3 and odd k, up to a term which depends on Kloosterman sums. In addition, applying simpler methods, we prove a theorem providing Niho type decimations with four-valued cross-correlation functions and their distribution. We conjecture that the latter result actually covers all such decimations.

On Niho type cross-correlation functions of m-sequences

Assume that d ≡ 1 ( mod q − 1 ) . We study the number of solutions to ( x + 1 ) d = x d + 1 in GF ( q 2 ) . In addition, we give a simple proof of the fact that the cross-correlation function between two m-sequences, which differ by a decimation of this type, is at least four-valued.

Construction of bent functions via Niho power functions

A Boolean function with an even number n = 2 k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions. Boolean functions of the form f ( x ) = tr ( α 1 x d 1 + α 2 x d 2 ) , α 1 , α 2 , x ∈ F 2 n , are considered, where the exponents d i ( i = 1 , 2 ) are of Niho type, i.e. the restriction of x d i on F 2 k is linear. We prove for several pairs of ( d 1 , d 2 ) that f is a bent function, when α 1 and α 2 fulfill certain conditions. To derive these results we develop a new method to prove that certain rational mappings on F 2 n are bijective.

Ternary m-sequences with three-valued cross-correlation function: new decimations of Welch and Niho type

We show that the cross correlation between two ternary m-sequences of period 3/sup n/-1 that differ by the decimation d=2/spl middot/3/sup m/+1, where n=2m+1, takes on three different values. We conjecture the same result for the decimation d=2/spl middot/3/sup r/+1, where n is odd and r is defined by the condition 4r+1/spl equiv/0 mod n. These two new cases form in a sense ternary counterparts of two previously confirmed binary cases, the conjectures of Welch and Niho (1972).