Highly Nonlinear Resilient Functions Optimizing Siegenthaler's Inequality

@inproceedings{Maitra1999HighlyNR,
  title={Highly Nonlinear Resilient Functions Optimizing Siegenthaler's Inequality},
  author={Subhamoy Maitra and Palash Sarkar},
  booktitle={CRYPTO},
  year={1999}
}
Siegenthaler proved that an n input 1 output, m-resilient (balanced mth order correlation immune) Boolean function with algebraic degree d satisfies the inequality : m + d ≤ n - 1. We provide a new construction method using a small set of recursive operations for a large class of highly nonlinear, resilient Boolean functions optimizing Siegenthaler's inequality m + d = n - 1. Comparisons to previous constructions show that better nonlinearity can be obtained by our method. In particular, we… 
On the Coset Weight Divisibility and Nonlinearity of Resilient and Correlation-Immune Functions
TLDR
A bound on the nonlinearity of resilient functions involving n, m and d is deduced, which improves upon those given recently and independently by Sarkar and Maitra and by Tarannikov and stands in the more general framework of m-th order correlation-immune functions.
Construction of Nonlinear Boolean Functions with Important Cryptographic Properties
TLDR
It is shown that for each positive integer m, there are infinitely many integers n, such that it is possible to construct n-variable, m-resilient functions having nonlinearity greater than 2n-1 -2[n/2].
On Boolean Functions with Low Polynomial Degree and Higher Order Sensitivity
TLDR
This paper connects the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial degree and high sensitivity and presents a construction with low (n − ω(1) order sensitivity exploiting Maiorana-McFarland constructions, that is borrowed from construction of resilient functions.
BDD based construction of resilient functions
TLDR
BDDs with attributed edges are made use of to provide an implementation of two construction meth- ods proposed by Maitra and Sakar and it is demonstrated that the size of BDDs of resilient functions obtained in this way grows linearly with the number of variables.
Construction of Highly Nonlinear Plateaued Resilient Functions with Disjoint Spectra
TLDR
The nonlinearity of the constructed functions (for some functions) has improved upon the bounds obtained by Gao et al., and some new constructions of highly nonlinear resilient Boolean functions on large number of variables with disjoint spectra by concatenating disjointed spectra functions on small number of variable.
On Resilient Boolean Functions with Maximal Possible Nonlinearity
It is proved that the maximal possible nonlinearity of n- variable m-resilient Boolean function is 2n-1-2m+1 for 2n-7/ 3 ≤ m ≤ n-2. This value can be achieved only for optimized functions (i. e.
On Nonlinearity and Autocorrelation Properties of Correlation Immune Boolean Functions
  • S. Maitra
  • Computer Science, Mathematics
    J. Inf. Sci. Eng.
  • 2004
TLDR
This paper provides a construction method for unbalanced, first order correlation immune Boolean functions on even an number of variables n ≥ 6, and points out the weakness of two recursive construction techniques for resilient functions in terms of autocorrelation values.
On Cryptographic Properties of the Cosets of
TLDR
A new approach for the study of weight distributions of cosets of the Reed-Muller code of order is introduced, based on the method introduced by Kasami in (1), using Pless identities to obtain a condition for a coset to have a "high" minimum weight.
Cryptographically significant Boolean functions with five valued Walsh spectra
...
...

References

SHOWING 1-10 OF 18 REFERENCES
Correlation-immunity of nonlinear combining functions for cryptographic applications
TLDR
A new class of combining functions is presented, which provides better security against correlation attacks, and the security is quantified by the smallest number m + 1 of subsequences that must be simultaneously considered in a correlation attack.
A spectral characterization of correlation-immune combining functions
It is shown that a Boolean combining function f(x) of n variables is mth-order correlation-immune if and only if its Walsh transform F( omega ) vanishes for all omega with Hamming weight between 1
Nonlinearity Criteria for Cryptographic Functions
TLDR
Nonlinearity criteria for Boolean functions are classified in view of their suitability for cryptographic design and two criteria turn out to be of special interest, the distance to linear structures and the Distance to affine functions, which are shown to be invariant under all affine transformations.
Decrypting a Class of Stream Ciphers Using Ciphertext Only
  • T. Siegenthaler
  • Computer Science, Mathematics
    IEEE Transactions on Computers
  • 1985
TLDR
The conclusion from the analysis is that the pseudonoise generator's output sequence and the sequences generated by the linear feedback shift registers should be uncorrelated, which leads to constraints for the nonlinear combining function to be used.
Products of linear recurring sequences with maximum complexity
TLDR
Conditions are derived which guarantee that products of linear recurring sequences attain maximum linear complexity, and results obtained are extended to arbitrary linear combinations of product sequences.
Highly Nonlinear Balanced Boolean Functions with a Good Correlation-Immunity
We study a corpus of particular Boolean functions: the idempotents. They enable us to construct functions which achieve the best possible tradeoffs between the cryptographic fundamental properties:
Heuristic Design of Cryptographically Strong Balanced Boolean Functions
TLDR
The definitions for some cryptographic properties are generalised, providing a measure suitable for use as a fitness function in a genetic algorithm seeking balanced Boolean functions that satisfy both correlation immunity and the strict avalanche criterion.
More Correlation-Immune and Resilient Functions over Galois Fields and Galois Rings
We show that the usual constructions of bent functions, when they are suitably modified, allow constructions of correlation-immune and resilient functions over Galois fields and, in some cases, over
Shift-register synthesis and BCH decoding
  • J. Massey
  • Computer Science
    IEEE Trans. Inf. Theory
  • 1969
It is shown in this paper that the iterative algorithm introduced by Berlekamp for decoding BCH codes actually provides a general solution to the problem of synthesizing the shortest linear feedback
On Correlation-Immune Functions
TLDR
A recursive definition of any correlation-immune function of maximal degree is given and the set of quadratic balanced correlation- immune functions of maximal order is described.
...
...