Perfect Nonlinear Functions Research Articles

Construction of optimal family of frequency-hopping sequences based on perfect nonlinear function

A new family of frequency-hopping sequences was constructed based on perfect nonlinear function.It was proved that these sequences had good Hamming auto-correlation and cross-correlation properties according to the properties of perfect nonlinear function.Based upon different perfect nonlinear functions,different frequency-hopping sequence families with optimal Hamming correlation property could be obtained.

Reversed Dickson polynomials over finite fields

Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D n ( x , a ) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on F q with q < 200 , with only one exception, all belong to the RDPP families established in this paper.

On the constructions of constant-composition codes from perfect nonlinear functions

A new construction of constant-composition codes based on all known perfect nonlinear functions from F q m to itself is presented, which provides a kind of unified constructions of constant-composition codes based on all known perfect nonlinear functions from F q m to itself. It is proved that the new constant-composition codes are optimal with respect to the Luo-Fu-Vinck-Chen bound, when m is an odd positive integer greater than 1. Finally, we point out that two constructions of constant-composition codes, proposed by Ding Cunsheng et al. in 2005, are equivalent to two special types of the new constant-composition codes.

New families of perfect nonlinear polynomial functions

A new class of almost perfect nonlinear (APN) polynomial functions has been recently introduced. We give some generalizations of these functions and deduce new families of perfect nonlinear (PN) functions. We show that these PN functions are CCZ-inequivalent to the known perfect nonlinear functions.

A new construction of perfect nonlinear functions using Galois rings

AbstractFor a prime p, we give a construction of perfect nonlinear functions from ℤ to ℤ when either of the following conditions holds: (1) n≥p; (2) n&lt;p, and n is a composite number or is the sum of positive composite numbers. It follows that when n≥12, there is a perfect nonlinear function from ℤ to ℤ for any prime p. © 2009 Wiley Periodicals, Inc. J Combin Designs 17: 229‐239, 2009

A new almost perfect nonlinear function which is not quadratic

Following an example in [12], we show how to change one coordinate function of an almost perfect nonlinear (APN) function in order to obtain new examples. It turns out that this is a very powerful method to construct new APN functions. In particular, we show that our approach can be used to construct a ''non-quadratic'' APN function. This new example is in remarkable contrast to all recently constructed functions which have all been quadratic. An equivalent function has been found independently by Brinkmann and Leander [8]. However, they claimed that their function is CCZ equivalent to a quadratic one. In this paper we give several reasons why this new function is not equivalent to a quadratic one.

Properties and Applications of Preimage Distributions of Perfect Nonlinear Functions

The preimage distributions of perfect nonlinear functions from an Abelian group of order n to an Abelian group of order 3 or 4, respectively, are studied. Based on the properties of the preimage distributions of perfect nonlinear functions from an Abelian group of order 3r to an Abelian group of order 3, the weight distributions of the ternary linear codes C Pi from the perfect nonlinear functions Pi(x) from F 3 r to itself are determined. These results suggest that two open problems, proposed by Carlet, Ding, and Yuan in 2005 and 2006, respectively, are answered.

On the Covering Structures of Two Classes of Linear Codes From Perfect Nonlinear Functions

In this paper, the weight distributions of two classes of linear codes based on all known explicit perfect nonlinear functions from <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Fqm</i> to itself are determined using a unified approach. All the minimal codewords of these codes are characterized according to their weights, which suggests that their covering structures are determined. Finally, all the minimal access sets of the secret sharing schemes based on their dual codes are obtained.

Fourier Spectra of Binomial APN Functions

In this paper we compute the Fourier spectra of some recently discovered binomial almost perfect nonlinear (APN) functions. One consequence of this is the determination of the nonlinearity of the functions, which measures their resistance to linear cryptanalysis. Another consequence is that certain error-correcting codes related to these functions have the same weight distribution as the 2-error-correcting Bose-Chaudury-Hocquenghem (BCH) code. Furthermore, for field extensions of $\mathbb{F}_2$ of odd degree, our results provide an alternative proof of the APN property of the functions.

Optimal Constant Composition Codes From Zero-Difference Balanced Functions

Constant composition codes are a special class of constant weight codes, and include permutation codes as a subclass. They have applications in communications engineering. In this correspondence, a generic construction of optimal constant composition codes using zero-difference balanced functions is introduced. It generalizes the earlier construction of optimal constant composition codes employing perfect nonlinear functions. In addition, two classes of optimal constant composition codes with new parameters are reported.

On perfect nonlinear functions (ź)

Perfect nonlinear functions are of importance in cryptography. By using Galois ring, relative trace and investigating the character values of corresponding relative difference sets, we present a co...

Direct sums of balanced functions, perfect nonlinear functions, and orthogonal cocycles

AbstractDetermining if a direct sum of functions inherits nonlinearity properties from its direct summands is a subtle problem. Here, we correct a statement by Nyberg on inheritance of balance and we use a connection between balanced derivatives and orthogonal cocycles to generalize Nyberg's result to orthogonal cocycles. We obtain a new search criterion for PN functions and orthogonal cocycles mapping to non‐cyclic abelian groups and use it to find all the orthogonal cocycles over Z2t, 2 ≤ t ≤ 4. We conjecture that any orthogonal cocycle over Z2t, t ≥ 2, must be multiplicative. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 173–181, 2008

On perfect nonlinear functions (Π)

Perfect nonlinear functions are of importance in cryptography. By using Galois ring, relative trace and investigating the character values of corresponding relative difference sets, we present a construction of perfect nonlinear functions from $${\mathbb{Z}_4^{2m}}$$ to $${\mathbb{Z}_4^{m'}}$$, where m′ is a divisor of 2m, and a construction of perfect nonlinear functions from $${\mathbb{Z}_{p^2}^{n}}$$ to $${\mathbb{Z}_{p^2}^m}$$ where 2m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from $${\mathbb{Z}_{2p}^2}$$ to $${\mathbb{Z}_{2p}}$$ if and only if p = 2, and there doesn’t exist a perfect nonlinear function from $${\mathbb{Z}_{2^kl}^{2n}}$$ to $${\mathbb{Z}_{2^kl}^m}$$ if m > n and l(l is odd) is self-conjugate modulo 2 k (k ≥ 1).

A New Criterion for Nonlinearity of Block Ciphers

For years, the cryptographic community has searched for good nonlinear functions. Bent functions, almost perfect nonlinear functions, and similar constructions have been suggested as a good base for cryptographic applications due to their highly nonlinear nature. In the first part of this paper, we examine using these functions as block ciphers, and present several distinguishers between almost perfect nonlinear permutations and random permutations. In the second part of the paper, we suggest a criterion to measure the effective linearity of a given block cipher. We devise a general distinguisher for block ciphers based on their effective linearity. Finally, we show that for several constructions, our distinguishing attack is better than previously known techniques.

G-Perfect nonlinear functions

Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replace...

Value Distributions of Exponential Sums From Perfect Nonlinear Functions and Their Applications

In this paper we present a unified way to determine the values and their multiplicities of the exponential sums SigmaxisinF(q)zetap Tr(af(x)+bx)(a,bisinFq,q=pm,pges3) for all perfect nonlinear functions f which is a Dembowski-Ostrom polynomial or p = 3,f=x(3(k)+1)/2 where k is odd and (k,m)=1. As applications, we determine (1) the correlation distribution of the m-sequence {alambda=Tr(gammalambda)}(lambda=0,1,...) and the sequence {blambda=Tr(f(gammalambda))}(lambda=0,1,...) over Fp where gamma is a primitive element of Fq and (2) the weight distributions of the linear codes over Fp defined by f.

Algebraic Constructions of Optimal Frequency-Hopping Sequences

Frequency-hopping (FH) spread spectrum and direct-sequence spread spectrum are two main spread-coding technologies. Frequency-hopping sequences are needed in FH code-division multiple-access (CDMA) systems. In this correspondence, three classes of optimal frequency-hopping sequences are constructed with algebraic methods. The three classes are based on perfect nonlinear functions, power functions, and norm functions, respectively. Both individual optimal frequency-hopping sequences and optimal families of frequency-hopping sequences are presented.

The weight distribution of a class of linear codes from perfect nonlinear functions

In this correspondence, the weight distribution of a class of linear codes based on perfect nonlinear functions (also called planar functions) is determined. The class of linear codes under study are either optimal or among the best codes known, and have nice applications in cryptography.

Quadratic Equations from APN Power Functions

We develop several tools to derive quadratic equations from algebraic S-boxes and to prove their linear independence. By applying them to all known almost perfect nonlinear (APN) power functions and the inverse function, we can estimate the resistance against algebraic attacks. As a result, we can show that APN functions have different resistance against algebraic attacks, and especially S-boxes with Gold or Kasami exponents have very weak resistance.

A family of skew Hadamard difference sets

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley–Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new perfect nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these perfect nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley–Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new perfect nonlinear functions has applications in cryptography, coding theory, and combinatorics.