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### p-ary sequences with six-valued cross-correlation function: a new decimation of Niho type [article]

Yuhua Sun, Hui Li, Zilong Wang, Tongjiang Yan
2011 arXiv   pre-print
Using generalized Niho's Theorem, we show that the cross-correlation function between a p-ary m-sequence of period p^n-1 and its decimated sequence by the above d is at most six-valued and we can easily  ...  For an odd prime p and n=2m, a new decimation d=(p^m-1)^2/2+1 of Niho type of m-sequences is presented.  ...  Conclusion In this note, using generalized Niho's Theorem, we give an alternative proof of a known result.  ...

### Binary m-sequences with three-valued crosscorrelation: a proof of Welch's conjecture

A. Canteaut, P. Charpin, H. Dobbertin
2000 IEEE Transactions on Information Theory
a binary maximum-length linear shift register sequences of degree and a decimation of that sequence by 2 + 3 takes on precisely the three values 1 1 2 +1 .  ...  We prove the long-standing conjecture of Welch stating that for odd = 2 + 1, the power function with = 2 + 3 is maximally nonlinear on GF (2 ) or, in other terms, that the crosscorrelation function between  ...  They would also like to thank the members of the Research Group ESAT-COSIC, especially Prof. B. Preneel, for their hospitality.  ...

### Construction of bent functions via Niho power functions

Hans Dobbertin, Gregor Leander, Anne Canteaut, Claude Carlet, Patrick Felke, Philippe Gaborit
2006 Journal of combinatorial theory. Series A
A Boolean function with an even number n = 2k of variables is called bent if it is maximally nonlinear. We present here a new construction of bent functions.  ...  We prove for several pairs of (d 1 , d 2 ) that f is a bent function, when 1 and 2 fulfill certain conditions.  ...  Niho's theorem and Dickson polynomials Niho's theorem  is presented below. For the reader's convenience, we include a proof (cf.  ). Theorem 10.  ...

### A New Characterization of Almost Bent Functions [chapter]

Anne Canteaut, Pascale Charpin, Hans Dobbertin
1999 Lecture Notes in Computer Science
We here give a necessary and sufficient condition for an almost perfect nonlinear function to be almost bent.  ...  This notably enables us to exhibit some infinite families of power functions which are not almost bent. L. Knudsen (Ed.): FSE'99  ...  Since R n is a principal domain, any cyclic code C of length n is generated by a unique monic polynomial g having minimal degree.  ...

### Page 4595 of Mathematical Reviews Vol. , Issue 2002F [page]

2002 Mathematical Reviews
Pless power moment identities and a theorem concerning cyclic codes due to McEliece are used to reduce the conjectures to theorems due to Dobbertin concerning certain almost perfect nonlinear mappings.  ...  In this note we generalize this to |1 + Cy(t)| < cel /pr for p > 3.  ...

### The resolution of Niho's last conjecture concerning sequences, codes, and Boolean functions [article]

Tor Helleseth, Daniel J. Katz, Chunlei Li
2021 arXiv   pre-print
A new method is used to resolve a long-standing conjecture of Niho concerning the crosscorrelation spectrum of a pair of maximum length linear recursive sequences of length 2^2 m-1 with relative decimation  ...  The method used to obtain this result proves constraints on the number of roots that certain seventh degree polynomials can have on the unit circle of a finite field.  ...  A generalization of Niho's result was stated in [Ros06, Theorem 2]; we now state and prove a corrected 3 version. Lemma 2.5.  ...

### On the Weight Distribution of Cyclic Codes with Niho Exponents [article]

Shuxing Li, Tao Feng, Gennian Ge
2014 arXiv   pre-print
More specifically, we obtain two classes of binary three-weight and four-weight cyclic codes and a class of nonbinary four-weight cyclic codes.  ...  Recently, there has been intensive research on the weight distributions of cyclic codes. In this paper, we compute the weight distributions of three classes of cyclic codes with Niho exponents.  ...  In the second part, we introduce Delsarte's Theorem and Niho's Theorem. A generalization of Niho's Theorem over odd characteristic is also presented.  ...

### Weight Divisibility of Cyclic Codes, Highly Nonlinear Functions on F2m, and Crosscorrelation of Maximum-Length Sequences

Anne Canteaut, Pascale Charpin, Hans Dobbertin
2000 SIAM Journal on Discrete Mathematics
Using McEliece's theorem we give some general results on the weight divisibility of duals of cyclic codes with two zeros; specifically, we exhibit some infinite families of pairs of maximum-length sequences  ...  Primitive cyclic codes with two zeros whose dual satisfies this property actually correspond to almost bent power functions and to pairs of maximum-length sequences with preferred crosscorrelation.  ...  This general condition generalizes the Sarwate-Pursley conjecture [31, 25] on the nonexistence of preferred pairs of m-sequences of length (2 m −1) when 4 divides m. Theorem 7.1.  ...

### New pairs of m-sequences with 4-level cross-correlation

Tor Helleseth, Petri Rosendahl
2005 Finite Fields and Their Applications
Let be a primitive element of GF (2 n ), where n ≡ 0(mod 4). Let d = (2 2k + 2 s+1 − 2 k+1 − 1)/(2 s −1), where n=2k, and s is such that 2s divides k.  ...  We prove that the binary m-sequences s(t) = tr( t ) and s(dt) have a four-level cross-correlation function and give the distribution of the values.  ...  A central problem in the theory of m-sequences, and in sequence design in general, is to determine the values and the number of occurrences of each value taken on by the cross-correlation function C d  ...

### On Niho type cross-correlation functions of m-sequences

Tor Helleseth, Jyrki Lahtonen, Petri Rosendahl
2007 Finite Fields and Their Applications
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.  ...  We study the number of solutions to (x + 1) d = x d + 1 in GF(q 2 ).  ...  Secondly, a family containing both of Niho's families was found in  . Niho's theorem in turn was generalized to nonbinary sequences in  .  ...

### On Inversion in Z_2^n-1 [article]

Gohar M. Kyureghyan, Valentin Suder
2013 arXiv   pre-print
We studied the function (n), which for a fixed positive integer d maps integers n≥ 1 to the least positive residue of the inverse of d modulo 2^n-1, if it exists.  ...  In this paper we determined explicitly the multiplicative inverses of the Dobbertin and Welch APN exponents in Z_2^n-1, and we described the binary weights of the inverses of the Gold and Kasami exponents  ...  For a generic integer d we cannot of course expect to be able to guess its inverse modulo 2 n − 1.  ...

### The only crooked power functions are x2k+2l

Gohar M. Kyureghyan
2007 European journal of combinatorics (Print)
} is the complement of a hyperplane for every fixed a ∈ F * 2 n (where F 2 n is considered as a vector space over F 2 ).  ...  Our proof is an easy generalization of the proof given in  . Given α ∈ F * 2 n , we define b α : F 2 n → F 2 by b α (β) = 1 if W f (α, β) = 0 0 otherwise. Lemma 2.  ...  Lemma 2 and Theorem 2 imply the following statement. Theorem 2 ([ |F α ∩ H i (β 1 ) ∩ H j (β 2 )| ∈ {2 n−3 , 2 n−3 ± 2 n−3 2 }, for all α, β 1 = β 2 ∈ F * 2 n and i, j ∈ F 2 .  ...

### On the Correlation Distribution for a Ternary Niho Decimation [article]

Yongbo Xia, Nian Li, Xiangyong Zeng, Tor Helleseth
2016 arXiv   pre-print
By studying the weight distribution of the ternary Zetterberg code and counting the numbers of solutions of some equations over the finite field F_3^n, the correlation distribution between a ternary m-sequence  ...  This is the first time that the correlation distribution for a non-binary Niho decimation has been determined since 1976.  ...  The possible values of S(a, b) given in Lemma 7 below can be found by the techniques used in Lemma 2 of  , which originate from the proof of Niho's Theorem  .  ...

### Crooked maps in F2n

Gohar M. Kyureghyan
2007 Finite Fields and Their Applications
This is a generalization of a result in  . There are some indications that the complete characterization of crooked maps is difficult.  ...  In Section 2 we generalize the notion of crooked maps and give some properties of such maps.  ...  A special case of Lemma 2 with p = 2, odd n and δ = γ = 1 was proved in [13, 18] . The statement of Corollary 6 can be generalized for the following class of maps. Proof.  ...

### Improved Lower Bounds for the Fourier Entropy/Influence Conjecture via Lexicographic Functions [article]

Rani Hod
2017 arXiv   pre-print
Additionally, we prove a Lipschitz-type condition on the total influence and spectral entropy, which may be of independent interest.  ...  The Fourier Entropy/Influence conjecture of Friedgut and Kalai from 1996 states that the entropy to influence ratio is bounded by a universal constant C.  ...  Generalizing the composition method Here is a sketch of the proof of Proposition 1.2, as done in [9, Lemma 5.1].  ...
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