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### 9-variable Boolean functions with nonlinearity 242 in the generalized rotation symmetric class

Selçuk Kavut, Melek Diker Yücel
2010 Information and Computation
To improve this nonlinearity result, we have firstly defined some subsets of the n-variable Boolean functions as the generalized classes of "k-RSBFs and k-DSBFs (k-Dihedral Symmetric Boolean Functions)  ...  In 2006, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions  ...  Gregor Leander's discussion on the linear equivalence of Boolean functions that are invariant under random permutations, which has helped us finding one missing class in Table 3 .  ...

### 9-variable Boolean Functions with Nonlinearity 242 in the Generalized Rotation Class [article]

Selcuk Kavut, Melek Diker Yucel
2008 arXiv   pre-print
To improve this nonlinearity result, we have firstly defined some subsets of the n-variable Boolean functions as the "generalized classes of k-RSBFs and k-DSBFs (k-Dihedral Symmetric Boolean Functions)  ...  In 2006, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions  ...  Gregor Leander for his discussion on linear equivalence of Boolean functions that are invariant under random permutations, which has helped us find one missing class in Table 3 .  ...

### New Patterson--Wiedemann type functions with 15 variables in the generalized rotation-symmetric class

Selçuk KAVUT
2017 Turkish Journal of Electrical Engineering and Computer Sciences
Recently, it was shown that there is no Boolean function on 15 variables with nonlinearity greater than 16276 in the class of functions that are invariant under the action of GF(2 3 ) * × GF (2 5 ) * .  ...  In this study, we consider some important subsets of this class and perform an efficient enumeration of the 15-variable Patterson-Wiedemann (PW) type functions with nonlinearity greater than the bent concatenation  ...  The existence of Boolean functions with nonlinearity exceeding the bent concatenation bound could be demonstrated for the first time for n = 15 by Patterson and Wiedemann [5] in 1983 using some combinatorial  ...

### Patterson–Wiedemann Type Functions on 21 Variables With Nonlinearity Greater Than Bent Concatenation Bound

Selcuk Kavut, Subhamoy Maitra
2016 IEEE Transactions on Information Theory
Some of these Boolean functions posses nonlinearity greater than the bent concatenation bound. The next possible option for exploring such functions is on 7 × 3 = 21 variables.  ...  Nonlinearity is one of the most challenging combinatorial property in the domain of Boolean function research.  ...  The support of the function f is invariant under the action of L * and J * .  ...

### Nonlinear Invariant Attack: Practical Attack on Full SCREAM, iSCREAM, and Midori64

Yosuke Todo, Gregor Leander, Yu Sasaki
2018 Journal of Cryptology
In this paper we introduce a new type of attack, called nonlinear invariant attack.  ...  Moreover, the nonlinear invariant attack on the underlying (tweakable) block cipher can be extended to a ciphertext-only attack in well-known modes of operation such as CBC or CTR.  ...  Finally, in Section 6, we give some additional insights into the general structure of nonlinear invariant functions and outline some future work.  ...

### Nonlinear Approximations in Cryptanalysis Revisited

Christof Beierle, Anne Canteaut, Gregor Leander
2018 IACR Transactions on Symmetric Cryptology
This work studies deterministic and non-deterministic nonlinear approximations for cryptanalysis of block ciphers and cryptographic permutations and embeds it into the well-understood framework of linear  ...  For a deterministic (i.e., with correlation ±1) nonlinear approximation we show that in many cases, such a nonlinear approximation implies the existence of a highly-biased linear approximation.  ...  We consider the attack strategy of approximating a (non-constant) Boolean function of the output by a Boolean function of the input.  ...

### Nonlinear Approximations in Cryptanalysis Revisited

Christof Beierle, Anne Canteaut, Gregor Leander
2018 IACR Transactions on Symmetric Cryptology
This work studies deterministic and non-deterministic nonlinear approximations for cryptanalysis of block ciphers and cryptographic permutations and embeds it into the well-understood framework of linear  ...  For a deterministic (i.e., with correlation ±1) nonlinear approximation we show that in many cases, such a nonlinear approximation implies the existence of a highly-biased linear approximation.  ...  We consider the attack strategy of approximating a (non-constant) Boolean function of the output by a Boolean function of the input.  ...

### On the Higher Order Nonlinearities of Algebraic Immune Functions [chapter]

Claude Carlet
2006 Lecture Notes in Computer Science
The values of the nonlinearity profile are known for very few functions and these functions have little cryptographic interest.  ...  This simple criterion is not always well adapted to the concrete situation in which Boolean functions are used in symmetric cryptography, since changing one or several output bits of a Boolean function  ...  In the nonlinear filter sub-model, the content of some of the flip-flops in a single (longer) LFSR constitute the input to a nonlinear Boolean function which produces the keystream.  ...

### On Higher-order Correlation Immunity and Higher Nonlinearity for a Class of Boolean Functions

Jinglian Huang, Zhuo Wang, Chunling Zhang
2016 DEStech Transactions on Engineering and Technology Research
Meanwhile, with the lowest algebraic degree annihilator of Boolean functions, we also derive the invariance of the nonlinearity of Boolean functions with higher correlation immunity, and prove the existence  ...  of a class of a Boolean function with higher nonlinearity.  ...  Acknowledgement This work is supported by National Natural Science Foundation of China (Grant No. 61262085).  ...

### Page 3347 of Mathematical Reviews Vol. , Issue 88f [page]

1988 Mathematical Reviews
The author presents algorithms for enumeration of some kinds of Boolean functions. These algorithms are used for data analysis.  ...  A set Q of Boolean functions is said to be an invariant class if for each function f € Q all functions which can be obtained from f by adding fictive variables, or by renaming variables, or by identification  ...

### Page 1607 of Mathematical Reviews Vol. 42, Issue 6 [page]

1971 Mathematical Reviews
An algorithm is described which generalizes the algorithms of successive eliminations for solving a Boolean equation, for solving a pseudo- Boolean equation and for minimizing a pseudo-Boolean function  ...  A second algorithm generalizes a method for optimizing a pseudo-Boolean function subject to a single pseudo-Boolean constraint of the form F=0. Several applications are described. G.  ...

### Generalized Nonlinear Invariant Attack and a New Design Criterion for Round Constants

Yongzhuang Wei, Tao Ye, Wenling Wu, Enes Pasalic
2018 IACR Transactions on Symmetric Cryptology
In this article, to eliminate the impact from the round constants, a generalized nonlinear invariant attack which uses a pair of constants in the input of nonlinear invariants is proposed.  ...  The considered variant of iSCREAM is however resistant against nonlinear invariant attack of Todo et al..  ...  Yongzhuang Wei and Tao Ye are supported in part by the National Key R&D Program of China (No. 2017YFB0802000) and in part by the Natural Science Foundation of China (Nos. 61572148, 61872103).  ...

### Generalized Nonlinear Invariant Attack and a New Design Criterion for Round Constants

Yongzhuang Wei, Tao Ye, Wenling Wu, Enes Pasalic
2018 IACR Transactions on Symmetric Cryptology
In this article, to eliminate the impact from the round constants, a generalized nonlinear invariant attack which uses a pair of constants in the input of nonlinear invariants is proposed.  ...  The considered variant of iSCREAM is however resistant against nonlinear invariant attack of Todo et al..  ...  Yongzhuang Wei and Tao Ye are supported in part by the National Key R&D Program of China (No. 2017YFB0802000) and in part by the Natural Science Foundation of China (Nos. 61572148, 61872103).  ...

### Page 9504 of Mathematical Reviews Vol. , Issue 2004k [page]

2004 Mathematical Reviews
Summary: “The paper discusses the problem of recognizing Boolean function linearity. A spectral method of the analysis of Boolean functions using the Walsh transform is described.  ...  These simple models allow a qualitative analysis of the nonlinear dynamics of the decoding algorithm.  ...

### Four Measures of Nonlinearity [chapter]

Joan Boyar, Magnus Find, René Peralta
2013 Lecture Notes in Computer Science
Cryptographic applications, such as hashing, block ciphers and stream ciphers, make use of functions which are simple by some criteria (such as circuit implementations), yet hard to invert almost everywhere  ...  We also present new connections between two of these measures. Additionally, we give a lower bound on the multiplicative complexity of collision-free functions.  ...  Acknowledgements We are grateful to Meltem Turan for pointing out some typos in an earlier version of this paper.  ...
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