Vector Boolean Functions Research Articles

A Note on the Confusion Coefficient of Boolean Functions

Low confusion coefficient values can make side-channel attacks harder for vector Boolean functions in Block cipher. In this paper, we give new results of confusion coefficient for f ⊞ g, f ⊡ g, f ⊕ g and fg for different Boolean functions f and g, respectively. And we deduce a relationship on the sum-of-squares of the confusion coefficient between one n-variable function and two (n - 1)-variable decomposition functions. Finally, we find that the confusion coefficient of vector Boolean functions is affine invariant.

Parallel Fast Walsh Transform Algorithm and Its Implementation with CUDA on GPUs

Abstract Some of the most important cryptographic characteristics of the Boolean and vector Boolean functions (nonlinearity, autocorrelation, differential uniformity) are connected with the Walsh spectrum. In this paper, we present several algorithms for computing the Walsh spectrum implemented in CUDA for parallel execution on GPU. They are based on the most popular sequential algorithm. The algorithms differ in the complexity of implementations, resources used, optimization strategies and techniques. In the end, we give some experimental results.

Algorithm 959

VBF is a collection of C++ classes designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This implementation uses the NTL library from Victor Shoup, adding new modules that call NTL functions and complement the existing ones, making it better suited to cryptography. The class representing a vector Boolean function can be initialized by several alternative types of data structures such as Truth Table, Trace Representation, and Algebraic Normal Form (ANF) , among others. The most relevant cryptographic criteria for both block and stream ciphers as well as for hash functions can be evaluated with VBF: it obtains the nonlinearity, linearity distance, algebraic degree, linear structures, and frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions are presented. Finally, three real applications of the library are described: the first one analyzes the KASUMI block cipher, the second one analyzes the Mini-AES cipher, and the third one finds Boolean functions with very high nonlinearity, a key property for robustness against linear attacks.

Об алгебраической иммунности векторных булевых функций

Об алгебраической иммунности векторных булевых функций

Об обратимости векторных булевых функций

Об обратимости векторных булевых функций

On an improved correlation analysis of stream ciphers using multi-output Boolean functions and the related generalized notion of nonlinearity

We investigate the security of $n$-bit to $m$-bit vectorial Boolean functions in stream ciphers. Such stream ciphers have higher throughput than those using single-bit output Boolean functions. However, as shown by Zhang and Chan at Crypto 2000, linear approximations based on composing the vector output with any Boolean functions have higher bias than those based onthe usual correlation attack. In this paper, we introduce a new approach for analyzing vector Boolean functions called generalized correlation analysis. It is based on approximate equations which are linear in the input $x$ but of freedegree in the output $z = F(x)$. The complexity for computing the generalized nonlinearity for this new attack is reduced from $2$2m×n+n to $2$2n. Based on experimental results, we show that the new generalized correlation attack gives linear approximation with much higher bias than the Zhang-Chan andusual correlation attack. We confirm this with a theoretical upper bound for generalized nonlinearity, which is much lower than for the unrestricted non-linearity (for Zhang-Chan's attack) and a fortiori for usual nonlinearity. We also prove a lower bound for generalized nonlinearity which allows us to construct vector Boolean functions with high generalized nonlinearity from bent and almost bent functions. We derive the generalized nonlinearity of some known secondary constructions for secure vector Boolean functions. Finally, we prove that if a vector Boolean function has high nonlinearity or even a highunrestricted nonlinearity, it cannot ensure that it will have high generalized nonlinearity.

Algebraic degree of the inverse of linearized polynomials

The inverse of linearized polynomials might be a good candidate of vector Boolean functions for cryptographic applications since it is a generalization of the inverse function that is widely used in cryptographic primitives. In Crypto 2001, a construction method of vector resilient functions was proposed using linearized polynomials and linear codes. Unfortunately, the analysis of the algebraic degree of the inverse of linearized polynomials was wrong. In this paper, we correct the inexact result. More precisely, we give the exact maximal algebraic degree and an upper bound of the minimal algebraic degree.

Indices to measure the cryptographic strength of S-box

The vector Boolean functions with good cryptographic requirements are used for designing S-boxes for block ciphers and combiner/filter functions for stream ciphers. The security strength of any crypto algorithm is dependent on the security strength of S-boxes used in the design and immunity to the cryptanalytic attacks. In order to compare the security strength of various crypto algorithms, in this paper, various indices, have been developed based on the cryptographic properties and parameters for immunity to cryptanalytic attacks.

A series of maximum subalgebras of direct products of algebras of finite-valued logics

We construct a series of maximum subalgebras of direct products of algebras of finite-valued logics, which are subdirect products of the corresponding Boolean algebras. We show that subdirect products make the largest contribution to the number of maximum subalgebras. The completeness problems is solved in the algebra of vector Boolean functions.

Vector Boolean Algebra and Calculus

In this paper, binary-vector Boolean algebra, a generalization of ordinary Boolean algebra [6], is introduced. In this algebra, every element is represented by a binary vector and in addition to ordinary AND, OR, and NOT operations, a new operation called the rotation operation which rotates (rightward or leftward) the components of a binary vector is introduced. Moreover, the NOT or COMPLEMENTATION operation is extended to a more general operation called the generalized complement which includes the total complement (ordinary complement), the null complement (no complement), and newly introduced partial complements. Because of this generalization, all axioms and theorems of ordinary Boolean algebra are generalized. In particular, it is shown that DeMorgan's theorem, Shannon's theorem, and the expansion theorem are generalized into more general forms which include their corresponding ordinary version as a special case. It is also shown that any multivalued logic truth table can be represented by a vector Boolean function. Three compact canonical (sum-of-products and product-of-sums) forms of this function are presented.