Sugata Gangopadhyay

Investigations on Bent and Negabent Functions via the Nega-Hadamard Transform

Parker considered a new type of discrete Fourier transform, called nega-Hadamard transform. We prove several results regarding its behavior on combinations of Boolean functions and use this theory to derive several results on negabentness (that is, flat nega-spectrum) of concatenations, and partially symmetric functions. We derive the upper bound ⌈n/2⌉ for the algebraic degree of a negabent function on n variables. Further, a characterization of bent-negabent functions is obtained within a subclass of the Maiorana-McFarland set. We develop a technique to construct bent-negabent Boolean functions by using complete mapping polynomials. Using this technique, we demonstrate that for each ℓ ≥ 2, there exist bent-negabent functions on n = 12ℓ variables with algebraic degree n/4 + 1 = 3ℓ + 1. It is also demonstrated that there exist bent-negabent functions on eight variables with algebraic degrees 2, 3, and 4. Simple proofs of several previously known facts are obtained as immediate consequences of our work.

Internal state recovery of grain-v1 employing normality order of the filter function

A novel technique for cryptanalysis of the stream cipher Grain-v1 is given. In a particular setting, the algorithms proposed in this study provide recovery of an internal state of Grain-v1 with the expected time complexity of only 2 54 table look-up operations employing a memory of dimension ~2 70 , assuming availability of 2 34 keystream sequences each of length 2 38 generated for different initial values, and the pre-processing time complexity of ~2 88 . These figures appear as significantly better in comparison with the previously reported ones. The proposed approach for cryptanalysis primarily depends on the order of normality of the employed Boolean function in Grain-v1. Accordingly, in addition to the security evaluation insights of Grain-v1, the results of this study are also an evidence of the cryptographic significance of the normality criteria of Boolean functions.

GENERIC CRYPTOGRAPHIC WEAKNESS OF k-NORMAL BOOLEAN FUNCTIONS IN CERTAIN STREAM CIPHERS AND CRYPTANALYSIS OF GRAIN-128

This paper considers security implications of k-normal Boolean functions when they are employed in certain stream ciphers. A generic algorithm is proposed for cryptanalysis of the considered class of stream ciphers based on a security weakness of k-normal Boolean functions. The proposed algorithm yields a framework for mounting cryptanalysis against particular stream ciphers within the considered class. Also, the proposed algorithm for cryptanalysis implies certain design guidelines for avoiding certain weak stream cipher constructions. A particular objective of this paper is security evaluation of stream cipher Grain-128 employing the developed generic algorithm. Contrary to the best known attacks against Grain-128 which provide complexity of a secret key recovery lower than exhaustive search only over a subset of secret keys which is just a fraction (up to 5%) of all possible secret keys, the cryptanalysis proposed in this paper provides significantly lower complexity than exhaustive search for any secret key. The proposed approach for cryptanalysis primarily depends on the order of normality of the employed Boolean function in Grain-128. Accordingly, in addition to the security evaluation insights of Grain-128, the results of this paper are also an evidence of the cryptographic significance of the normality criteria of Boolean functions.

Nega-hadamard transform, bent and Negabent functions

In this paper we start developing a detailed theory of nega-Hadamard transforms. Consequently, we derive several results on ne-gabentness of concatenations, and partially-symmetric functions. We also obtain a characterization of bent-negabent functions in a subclass of Maiorana-McFarland set. As a by-product of our results we obtain simple proofs of several existing facts.

A Note on Generalized Bent Criteria for Boolean Functions

In this paper, we consider the spectra of Boolean functions with respect to the action of unitary transforms obtained by taking tensor products of the Hadamard kernel, denoted by <i>H</i>, and the nega-Hadamard kernel, denoted by <i>N</i>. The set of all such transforms is denoted by {<i>H</i>, <i>N</i>}ⁿ. A Boolean function is said to be bent₄ if its spectrum with respect to at least one unitary transform in {<i>H</i>, <i>N</i>}ⁿ is flat. We obtain a relationship between bent, semibent, and bent₄ functions, which is a generalization of the relationship between bent and negabent Boolean functions proved by Parker and Pott [cf., LNCS 4893 (2007), 9-23]. As a corollary to this result, we prove that the maximum possible algebraic degree of a bent₄ function on <i>n</i> variables is [<i>n</i>/2] and, hence, solve an open problem posed by Riera and Parker [cf., IEEE-TIT 52:9 (2006), 4142-4159].

Third-order nonlinearities of a subclass of Kasami functions

The rth-order nonlinearity, where r ≥ 1, of an n-variable Boolean function f, denoted by nlr(f), is defined as the minimum Hamming distance of f from all n-variable Boolean functions of degrees at most r. In this paper we obtain a lower bound of the third-order nonlinearities of Kasami functions of the form

Patterson-Wiedemann construction revisited

Tr_{1}^{n}(\mu x^{57})

A Lower Bound of the Second-order Nonlinearities of Boolean Bent Functions

. It is demonstrated that for large values of n the lower bound of the third-order nonlinearities of the functions of this form is larger than the general lower bound obtained by Carlet (IEEE Trans Inf Theory 54(3):1262–1272, 2008) for Kasami functions. Further we show that our result along with the computational results obtained by Fourquet and Tavernier (Designs Codes Cryptogr 49:323–340, 2008) provide us an estimate of the nonlinearity profiles of these functions for n = 7, 8, 10.

An Analysis of the Class of Bent Functions

In 1983, Patterson and Wiedemann constructed Boolean functions on n=15 input variables having nonlinearity strictly greater than 2^n^-^1-2^(^n^-^1^)^/^2. Construction of Boolean functions on odd number of variables with such high nonlinearity was not known earlier and also till date no other construction method of such functions are known. We note that the Patterson-Wiedemann construction can be understood in terms of interleaved sequences as introduced by Gong in 1995 and subsequently these functions can be described as repetitions of a particular binary string. As example we elaborate the cases for n=15,21. Under this framework, we map the problem of finding Patterson-Wiedemann functions into a problem of solving a system of linear inequalities over the set of integers and provide proper reasoning about the choice of the orbits. This, in turn, reduces the search space. Similar analysis also reduces the complexity of calculating autocorrelation and generalized nonlinearity for such functions. In an attempt to understand the above construction from the group theoretic view point, we characterize the group of all GF(2)-linear transformations of GF(2^a^b) which acts on PG(2,2^a).

Affine inequivalence of cubic Maiorana-McFarland type bent functions

In this paper we find a lower bound of the second-order nonlinearities of Boolean bent functions of the form