Recently, a framework for employing the Gowers U-2 norm in the context of (generalized) booleanfunctions with cryptographic significance was introduced. In this paper, we first give tight bounds on the Gowers U-2 nor...
详细信息
Recently, a framework for employing the Gowers U-2 norm in the context of (generalized) booleanfunctions with cryptographic significance was introduced. In this paper, we first give tight bounds on the Gowers U-2 norm of generalized boolean functions via the (generalized) sum-of-squares indicator. Secondly, we provide a framework for the generalized signal-to-noise ratio (GSNR) of generalized (n, m)-functions. We characterize the GSNR in terms of the Gowers U-2 norm. In particular, we present a direct link between the GSNR of a class of generalized boolean functions and the SNR of its component booleanfunctions. Finally, the expressions of the Gowers U-2 norm of generalized boolean functions from some well-known secondary constructions (the concatenation and Carlet's construction) are obtained.
We explicitly derive a connection between quantum circuits utilising IBM's quantum gate set and multivariate quadratic polynomials over integers modulo 8. We demonstrate that the action of a quantum circuit over i...
详细信息
ISBN:
(纸本)9781538659069
We explicitly derive a connection between quantum circuits utilising IBM's quantum gate set and multivariate quadratic polynomials over integers modulo 8. We demonstrate that the action of a quantum circuit over input qubits can be written as generalized Walsh-Hadamard transform. Here, we derive the polynomials corresponding to implementations of the Swap gate and Toffoli gate using IBM-Q gate set.
In this paper we investigate generalized boolean functions whose spectrum is flat with respect to a set of Walsh-Hadamard transforms defined using various complex primitive roots of 1. We also study some differential ...
详细信息
In this paper we investigate generalized boolean functions whose spectrum is flat with respect to a set of Walsh-Hadamard transforms defined using various complex primitive roots of 1. We also study some differential properties of the generalized boolean functions in even dimension defined in terms of these different characters. We show that those functions have similar properties to the vectorial bent functions. We next clarify the case of gbent functions in odd dimension. As a by-product of our proofs, more generally, we also provide several results about plateaued functions. Furthermore, we find characterizations of plateaued functions with respect to different characters in terms of second derivatives and fourth moments.
This dissertation investigates correlation immunity, avalanche features, and the bent cryp- tographic properties for generalized boolean functions defined on V n with values in Z q. We extend the concept of correlatio...
详细信息
This dissertation investigates correlation immunity, avalanche features, and the bent cryp- tographic properties for generalized boolean functions defined on V n with values in Z q. We extend the concept of correlation immunity from the boolean case to the generalized setting, and provide multiple construction methods for order 1 and higher correlation im- mune generalized boolean functions. We establish necessary and sufficient conditions for generalized boolean functions. Additionally, we discuss correlation immune and rotation symmetric generalized boolean functions, introducing a construction method along the way. Using a graph-theoretic and probabilistic frame of reference, we subsequently es- tablish several, increasingly stringent, strict avalanche criteria along with a construction method for generalized boolean functions. We introduce the notion of a uniform avalanche criterion and demonstrate that generalized boolean functions that satisfy this criterion are also order 1 correlation immune and always have boolean function components that are both order 1 correlation immune and satisfy the strict avalanche criterion. We subsequently investigate linear structures, directional derivatives and define a unit vector gradient for generalizedboolean function. We introduce the Walsh-Hadamard transform of a general- ized boolean function along with the notion of generalized bent booleanfunctions. We provide a construction of generalized bent booleanfunctions with outputs in Z 8 and estab- lish necessary conditions for generalized bent booleanfunctions.
In this paper, we investigate the properties of generalized bent functions defined on with values in , where q a parts per thousand yen 2 is any positive integer. We characterize the class of generalized bent function...
详细信息
In this paper, we investigate the properties of generalized bent functions defined on with values in , where q a parts per thousand yen 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana-McFarland type bent functions and Dillon's functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana-McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent booleanfunctions defined on with values in and . Moreover, we propose several constructions of such generalized bent functions for both n even and n odd.
In this paper, a new direct construction of zero correlation zone complementary sequence sets (ZCZ-CSs) via generalized boolean functions is proposed. The resultant ZCZ-CSs can meet the theoretical upper bound on the ...
详细信息
ISBN:
(纸本)9781728109626
In this paper, a new direct construction of zero correlation zone complementary sequence sets (ZCZ-CSs) via generalized boolean functions is proposed. The resultant ZCZ-CSs can meet the theoretical upper bound on the set size and have good peak-to-average power ratios (PAPRs). boolean monomials of degrees more than 2 are considered in our construction, so more ZCZ-CSs can be obtained. The larger number of ZCZ-CSs constructed by our method will increase possible applications in practical communication systems.
In this article we present a broader theoretical framework useful in studying the properties of so-called generalized bent functions. We give the sufficient conditions (and in many cases also necessary) for generalize...
详细信息
In this article we present a broader theoretical framework useful in studying the properties of so-called generalized bent functions. We give the sufficient conditions (and in many cases also necessary) for generalized bent functions when these functions are represented as a linear combination of: generalized bent;boolean bent;and a mixture of generalized bent and boolean bent functions. These conditions are relatively easy to satisfy and by varying the variables that specify these linear combinations many different classes of generalized bent functions can be derived. In particular, based on these results, we provide some generic construction methods of these functions and demonstrate that some previous methods are just special cases of the results given in this article.
The present paper introduces improvements to an algorithm for the minimization of generalized boolean functions presented in [3]. This algorithm employs a local covering technique, already used in an algorithm for the...
详细信息
暂无评论