Order Of Correlation Immunity Research Articles

On Unbalanced Boolean Functions with Best Correlation Immunity

It is known that the order of correlation immunity of a nonconstant unbalanced Boolean function in $n$ variables cannot exceed $2n/3-1$; moreover, it is $2n/3-1$ if and only if the function corresponds to an equitable $2$-partition of the $n$-cube with an eigenvalue $-n/3$ of the quotient matrix. The known series of such functions have proportion $1:3$, $3:5$, or $7:9$ of the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean function attains the correlation-immunity bound and has ratio $C:B$ of the number of ones and zeros, then $CB$ is divisible by $3$. In particular, this proves the nonexistence of equitable partitions for an infinite series of putative quotient matrices. 
 We also establish that there are exactly $2$ equivalence classes of the equitable partitions of the $12$-cube with quotient matrix $[[3,9],[7,5]]$ and $16$ classes, with $[[0,12],[4,8]]$. These parameters correspond to the Boolean functions in $12$ variables with correlation immunity $7$ and proportion $7:9$ and $1:3$, respectively (the case $3:5$ remains unsolved). This also implies the characterization of the orthogonal arrays OA$(1024,12,2,7)$ and OA$(512,11,2,6)$.

On Relationship of some Cryptographic Properties for Weight Symmetric H Boolean Functions

Using the derivative of the Boolean function and the e-derivative defined by ourselves as research tools, we go deep into the internal structure of the Boolean function values,and discuss the relationship of a variety of cryptographic properties of the weight symmetric H Boolean functions in the range of the weight with the existence of H Boolean functions. We get the results of the weight symmetric H Boolean function should have the same algebraic immunity order, correlation immunity order, the degree of diffusion and nonlinearity. The results provide a theoretical basis to reduce nearly half workload for studying the cryptographic properties of H Boolean function, and provides a new research methods for the study of the properties of cryptographic security of Boolean functions.

Relationship between Algebraic Immunity, Correlation Immune and Propagation of Boolean Functions

Using the derivative of the Boolean function and thee-derivative defined by ourselves as research tools, we study the problem of relationship between algebraic immunity,correlation immunity and propagation of H Boolean functions with weight of and satisfying the 1st-order propagation criterion togetherwith the problem of their compatibility. We get the results , suchas the relationship between the number of annihilators and correlation immunityorder, the relationship between the number of correlation immunity order and algebraic immune degree together with theircompatibility and the largest propagations of H Boolean function, the relationships between propagationsof Boolean function, correlation immunity order and algebraic immune degree.

Vectorial Boolean functions and induced algebraic equations

A general mathematical framework behind algebraic cryptanalytic attacks is developed. The framework relates to finding algebraic equations induced by vectorial Boolean functions and, in particular, equations of low algebraic degree. The equations may involve only a subset of input variables and may or may not be conditioned on the values of output variables. In addition, the equations may have a constrained form interesting for the so-called fast algebraic attacks. A possible divide-and-conquer effect is pointed out and the notion of algebraic immunity order, naturally extending the notion of correlation immunity order, is defined. An application of general results to stream ciphers known as combiners with or without memory, with possibly multiple outputs, is studied in particular detail and the concept of divide-and-conquer algebraic attacks is introduced. Special properties of combiners with finite input memory, such as nonlinear filter generators, are also established. It is also pointed out that Groumlbner basis algorithms may be used for finding low-degree induced algebraic equations

고속 병렬형 LM 이진 수열 발생기

LM 발생기는 합산 수열 발생기가 갖고 있는 최대 주기, 최대 근사 선형복잡도 및 최고 차수 상관면역도를 동시에 만족하는 발생기이며, 입출력 무상관 발생기이다. 본 논문에서는 LM 발생기를 고속화 구현하기 위하여 m-병렬 LM 수열 발생기(m-parallel LM-BSG, Lee-Moon Binary Sequence Generator)를 제안하였으며, m＝8인 병렬 발생기(8-parallel LM-BSG)를 세부 설계한다. 기존의 스트림 암호와 비교할 때, 제안된 알고리듬은 같은 안전성을 유지하면서 속도가 m＝8배 빨라짐을 알 수 있었다. The LM generator is an improved summation generator with maximum period, near maximum linear complexity and maximum order of correlation immunity, and it has a property with the input-output correlation immunity In this paper, we propose the high-speed m-parallel LM-BSG and 8-parallel LM-BSG for detail as a design example. When compared with a conventional stream cipher, the properties of the proposed cipher exhibited the same crypto-degree (security) with a rt times faster processing.

Connections among nonlinearity, avalanche and correlation immunity

Nonlinear Boolean functions play an important role in the design of block ciphers, stream ciphers and one-way hash functions. Over the years researchers have identified a number of indicators that forecast nonlinear properties of these functions. Studying the relationships among these indicators has been an area that has received extensive research. The focus of this paper is on the interplay of three notable nonlinear indicators, namely nonlinearity, avalanche and correlation immunity. We establish, for the first time, an explicit and simple lower bound on the nonlinearity N f of a Boolean function f of n variables satisfying the avalanche criterion of degree p, namely, N f ⩾2 n−1 −2 n−1−(1/2) p . We also identify all the functions whose nonlinearity attains the lower bound. As a further contribution of this paper, we prove that except for very few cases, the sum of the degree of avalanche and the order of correlation immunity of a Boolean function of n variables is at most n−2. The new results obtained in this work further highlight the significance of the fact that while avalanche property is in harmony with nonlinearity, both go against correlation immunity.

A note on the spectral characterization of correlation immune Boolean functions

We present a new proof of the Walsh transform characterization of correlation immune Boolean functions. Also we provide a simpler proof of the fundamental relation between order of correlation immunity and algebraic degree of a Boolean function.

On an improved summation generator with 2-bit memory

The summation generator is a real adder generator with a maximum period, near-maximum linear complexity and maximum order of correlation immunity. However it is neither secure against and nor immune to correlation attack between its output sequences and carry sequences in special cases. A modified summation generator, secure against such an attack, has recently been proposed, but no proof is given about its period and linear complexity. In this paper, we propose a new modified summation generator immune to correlation attack. Moreover, we conclude that its period is maximum and that its linear complexity is the same as that of the original summation generator.

Correlation-immune functions over finite fields

We give a series of constructions of correlation-immune function over finite fields. We prove that F/sub 2/ and F/sub 3/ are the only finite fields F/sub q/ with the property that every (n-1)th correlation-immune function in n>2 variables over F/sub q/ is linear. We also show that by choosing larger finite fields one can alleviate the tradeoff between the length of the linear equivalent and the order of correlation immunity. This is useful for the design of various cryptosystems.

Spectral characterization of the correlation-immunity of multiple-valued logic functions

Correlation-immunity is an important concept in cryptology. In ref. [1] Siegenthaler introduced the mathematical definition of correlation-immunity and used the correlation-immunity order of logic functions as a measure for cipher systems to defend against correlation attacks. In terms of Walsh transform, the correlation immunity of binary-valued logic functions, i.e. Boolean functions, was studied in ref. [2], and a necessary

Entropy immunity of feedforward networks

Siegenthaler’s “Correlation Immunity” concept has been improved in this paper. From practical points of view, the new concept is more powerful than the original one in avoiding the trade-off between “the order of correlation immunity” and “the linear complexity” of key streams in cipher system. Bent functions are also introduced into the studies of linear approximation and entropy immunity for feedforward networks. New results and new methods are presented also.