Nonlinear Resilient Functions Research Articles

Large Sets of Disjoint Spectra Plateaued Functions Inequivalent to Partially Linear Functions

In this paper, we give an efficient method for constructing a large set of disjoint spectra functions without linear structures, which are not equivalent to partially linear functions. This positively answers the open problem [“how to construct a large set of disjoint spectra functions which are not (linearly equivalent to) partially linear functions” raised by Zhang and Xiao]. At the same time, this significantly extends a recent result of Zhang, where a method of specifying four disjoint spectra functions was given. It is demonstrated that such sets can be utilized in the design of highly nonlinear resilient functions. In the second part, based on a generalization of the indirect sum method, we give an alternative approach for designing sets of disjoint spectra functions of even larger cardinality than already given ones, but these functions then admit linear structures. In addition, it is shown that using suitable initial functions in the generalized indirect sum method we can specify highly nonlinear resilient Boolean functions (in odd number of input variables $n$ ) whose nonlinearity in many cases exceeds the current best known values. Moreover, we design some balanced functions (for odd $n$ ) that also achieve the highest nonlinearity known.

Generating highly nonlinear resilient Boolean functions resistance against algebraic and fast algebraic attacks

ABSTRACTBoolean functions play an important role in the design of stream ciphers. In this paper, a simulated annealing algorithm is designed to obtain Boolean functions satisfying all the needed criteria: high nonlinearity, 1‐resiliency, optimal algebraic degree, optimal (or suboptimal) algebraic immunity, and good immunity to fast algebraic attacks. These functions provide a good trade‐off among the criteria to resist the known cryptanalytic techniques. Copyright © 2014 John Wiley & Sons, Ltd.

Construction of highly nonlinear resilient Boolean functions satisfying strict avalanche criterion

Boolean functions with high nonlinearity, high resiliency and strict avalanche criterion (SAC) play an important role in the designs of conventional cryptographic systems. In this paper, a method is proposed to construct resilient Boolean functions on n variables (n even) satisfying SAC with nonlinearity > 2n−1 − 2n/2. A large class of cryptographic Boolean functions that were not known earlier were obtained.

Highly Nonlinear Resilient Functions without Linear Structures

Highly Nonlinear Resilient Functions without Linear Structures

A recursive construction of highly nonlinear resilient vectorial functions

Vectorial functions which map n-bits to m-bits are important cryptographic sub-primitives in the design of additive stream ciphers. In this paper, we provide a recursive construction of highly nonlinear resilient vectorial functions. For the first time, some new classes of almost optimal resilient vectorial functions are obtained. We also compare the nonlinearity obtained by our construction with the nonlinearity of previously known construction.

Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any m, one can construct infinitely many re-variable (n even), m-resilient functions with nonlinearity > 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sup> - 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-2</sup> . A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.

Maiorana–McFarland Class: Degree Optimization and Algebraic Properties

In this paper, we consider a subclass of the Maiorana-McFarland class used in the design of resilient nonlinear Boolean functions. We show that these functions allow a simple modification so that resilient Boolean functions of maximum algebraic degree may be generated instead of suboptimized degree in the original class. Preserving a high-nonlinearity value immanent to the original construction method, together with the degree optimization gives in many cases functions with cryptographic properties superior to all previously known construction methods. This approach is then used to increase the algebraic degree of functions in the extended Maiorana-McFarland (MM) class (nonlinear resilient functions F:GF(2)n |rarrGF(2)m derived from linear codes). We also show that in the Boolean case, the same subclass seems not to have an optimized algebraic immunity, hence not providing a maximum resistance against algebraic attacks. A theoretical analysis of the algebraic properties of extended Maiorana-McFarland class indicates that this class of functions should be avoided as a filtering function in nonlinear combining generators

Algebraic immunity for cryptographically significant Boolean functions: analysis and construction

Recently, algebraic attacks have received a lot of attention in the cryptographic literature. It has been observed that a Boolean function f used as a cryptographic primitive, and interpreted as a multivariate polynomial over F/sub 2/, should not have low degree multiples obtained by multiplication with low degree nonzero functions. In this paper, we show that a Boolean function having low nonlinearity is (also) weak against algebraic attacks, and we extend this result to higher order nonlinearities. Next, we present enumeration results on linearly independent annihilators. We also study certain classes of highly nonlinear resilient Boolean functions for their algebraic immunity. We identify that functions having low-degree subfunctions are weak in terms of algebraic immunity, and we analyze some existing constructions from this viewpoint. Further, we present a construction method to generate Boolean functions on n variables with highest possible algebraic immunity /spl lceil/n/2/spl rceil/ (this construction, first presented at the 2005 Workshop on Fast Software Encryption (FSE 2005), has been the first one producing such functions). These functions are obtained through a doubly indexed recursive relation. We calculate their Hamming weights and deduce their nonlinearities; we show that they have very high algebraic degrees. We express them as the sums of two functions which can be obtained from simple symmetric functions by a transformation which can be implemented with an algorithm whose complexity is linear in the number of variables. We deduce a very fast way of computing the output to these functions, given their input.

Highly Nonlinear Resilient Functions Through Disjoint Codes in Projective Spaces

Functions which map n-bits to m-bits are important cryptographic sub-primitives in the design of additive stream ciphers. We construct highly nonlinear t-resilient such functions ((n, m, t) functions) by using a class of binary disjoint codes, a construction which was introduced in IEEE Trans. Inform. Theory, Vol. IT-49 (2) (2003). Our main contribution concerns the generation of suitable sets of such disjoint codes. We propose a deterministic method for finding disjoint codes of length ? m by considering the points of PG $$(v-1, \mathbb{F}_{2^{m}}$$ ). We then obtain some lower bounds on the number of disjoint codes, by fixing some parameters. Through these sets, we deduce in certain cases the existence of resilient functions with very high nonlinearity values. We show how, thanks to our method, the degree and the differential properties of (n, m, t) functions can be improved.

Improved Construction of Nonlinear Resilient S-Boxes

We provide two new construction methods for nonlinear resilient functions. The first method is a simple modification of a construction due to Zhang and Zheng and constructs n-input, m-output resilient S-boxes with degree d>m. We prove by an application of the Griesmer bound for linear error-correcting codes that the modified Zhang-Zheng construction is superior to the previous method of Cheon in Crypto 2001. Our second construction uses a sharpened version of the Maiorana-McFarland technique to construct nonlinear resilient functions. The nonlinearity obtained by our second construction is better than previously known construction methods.

On construction of a class of nonlinear resilient functions

In this paper the decomposition formula of Walsh spectrum of boolean functions is used to construct a class of nonlinear resilient functions.

Cryptographically resilient functions

This correspondence studies resilient functions which have applications in fault-tolerant distributed computing, quantum cryptographic key distribution, and random sequence generation for stream ciphers. We present a number of new methods for synthesizing resilient functions. An interesting aspect of these methods is that they are applicable both to linear and nonlinear resilient functions. Our second major contribution is to show that every linear resilient function can be transformed into a large number of nonlinear resilient functions with the same parameters. As a result, we obtain resilient functions that are highly nonlinear and have a high algebraic degree.

An infinite class of counterexamples to a conjecture concerning nonlinear resilient functions

The main construction for resilient functions uses linear errorcorrecting codes; a resilient function constructed in this way is said to be linear. It has been conjectured that if a resilient function exists, then a linear function with the same parameters exists. In this note we construct infinite classes of nonlinear resilient functions from the Kerdock and Preparata codes. We also show that linear resilient functions having the same parameters as the functions that we construct from the Kerdock codes do not exist. Thus, the aforementioned conjecture is disproved.k