239 research outputs found
Asymptotic enumeration of correlation-immune boolean functions
A boolean function of boolean variables is {correlation-immune} of order
if the function value is uncorrelated with the values of any of the
arguments. Such functions are of considerable interest due to their
cryptographic properties, and are also related to the orthogonal arrays of
statistics and the balanced hypercube colourings of combinatorics. The {weight}
of a boolean function is the number of argument values that produce a function
value of 1. If this is exactly half the argument values, that is,
values, a correlation-immune function is called {resilient}.
An asymptotic estimate of the number of -variable
correlation-immune boolean functions of order was obtained in 1992 by
Denisov for constant . Denisov repudiated that estimate in 2000, but we will
show that the repudiation was a mistake.
The main contribution of this paper is an asymptotic estimate of
which holds if increases with within generous limits and specialises to
functions with a given weight, including the resilient functions. In the case
of , our estimates are valid for all weights.Comment: 18 page
Balanced Symmetric Functions over
Under mild conditions on , we give a lower bound on the number of
-variable balanced symmetric polynomials over finite fields , where
is a prime number. The existence of nonlinear balanced symmetric
polynomials is an immediate corollary of this bound. Furthermore, we conjecture
that are the only nonlinear balanced elementary symmetric
polynomials over GF(2), where , and we prove various results in support of this conjecture.Comment: 21 page
A Secure Random Number Generator with Immunity and Propagation Characteristics for Cryptography Functions
Cryptographic algorithms and functions should possess some of the important functional requirements such as: non-linearity, resiliency, propagation and immunity. Several previous studies were executed to analyze these characteristics of the cryptographic functions specifically for Boolean and symmetric functions. Randomness is a requirement in present cryptographic algorithms and therefore, Symmetric Random Function Generator (SRFG) has been developed. In this paper, we have analysed SRFG based on propagation feature and immunity. Moreover, NIST recommended statistical suite has been tested on SRFG outputs. The test values show that SRFG possess some of the useful randomness properties for cryptographic applications such as individual frequency in a sequence and block-based frequency, long run of sequences, oscillations from 0 to 1 or vice-versa, patterns of bits, gap bits between two patterns, and overlapping block bits. We also analyze the comparison of SRFG and some existing random number generators. We observe that SRFG is efficient for cryptographic operations in terms of propagation and immunity features
Counting and characterising functions with “fast points” for differential attacks
Higher order derivatives have been introduced by Lai in a cryptographic context.
A number of attacks such as differential cryptanalysis, the cube and the AIDA attack
have been reformulated using higher order derivatives. Duan and Lai have introduced the
notion of “fast points” of a polynomial function f as being vectors a so that computing the
derivative with respect to a decreases the total degree of f by more than one. This notion
is motivated by the fact that most of the attacks become more efficient if they use fast
points. Duan and Lai gave a characterisation of fast points and Duan et al. gave some results
regarding the number of functions with fast points in some particular cases. We firstly give
an alternative characterisation of fast points and secondly give an explicit formula for the
number of functions with fast points for any given degree and number of variables, thus
covering all the cases left open in Duan et al. Our main tool is an invertible linear change of
coordinates which transforms the higher order derivative with respect to an arbitrary set of
linearly independent vectors into the higher order derivative with respect to a set of vectors
in the canonical basis. Finally we discuss the cryptographic significance of our results
Matriochka symmetric Boolean functions
International audienceWe present the properties of a new class of Boolean functions defined as the sum of m symmetric functions with decreasing number of variables and degrees. The choice of this construction is justified by the possibility to study these functions by using tools existing for symmetric functions. On the one hand we show that the synthesis is well understood and give an upper bound on the gate complexity. On the other hand, we investigate the Walsh spectrum of the sum of two functions and get explicit formulae for the case of degree at most three
A Hybrid Approach to Formal Verification of Higher-Order Masked Arithmetic Programs
Side-channel attacks, which are capable of breaking secrecy via side-channel
information, pose a growing threat to the implementation of cryptographic
algorithms. Masking is an effective countermeasure against side-channel attacks
by removing the statistical dependence between secrecy and power consumption
via randomization. However, designing efficient and effective masked
implementations turns out to be an error-prone task. Current techniques for
verifying whether masked programs are secure are limited in their applicability
and accuracy, especially when they are applied. To bridge this gap, in this
article, we first propose a sound type system, equipped with an efficient type
inference algorithm, for verifying masked arithmetic programs against
higher-order attacks. We then give novel model-counting based and
pattern-matching based methods which are able to precisely determine whether
the potential leaky observable sets detected by the type system are genuine or
simply spurious. We evaluate our approach on various implementations of
arithmetic cryptographicprograms.The experiments confirm that our approach out
performs the state-of-the-art base lines in terms of applicability, accuracy
and efficiency
1-Resilient Boolean Functions on Even Variables with Almost Perfect Algebraic Immunity
Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions
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