On a method of synthesis of correlation-immune Boolean functions
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Ekaterina K. Karelina
Abstract
An efficient recursive method for synthesis of correlation-immune Boolean functions is proposed. At the first stage, this method uses minimal correlation-immune functions. A classification of 6-variable minimal correlation-immune functions under the Jevons group is put forward. New results on minimal correlation-immune functions are given.
Note: Originally published in Diskretnaya Matematika (2018) 30, №4, 12–28 (in Russian).
Funding: This research was carried out with the financial support of the Russian Foundation for Basic Research (grant no. 16-01-00470-A).
Acknowledgement
The author is deeply indebted to her scientific supervisors O. A. Logachev and E. K. Alekseev for posing the problem, for permanent attention to the work, and their assistance in preparation of the text of this paper.
References
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9 Appendix
Applying the above method of adding new variables to known minimal CI-functions of a small number of variables, we have succeeded in constructing a 4-stable function f0 of 8 variables. This function expands in a sum of 11 minimal CI-functions of 8 variables of weight 10, one minimal CI-function of 8 variables of weight 8, 1 minimal CI-function of 8 variables of weight 6, and 2 minimal CI-functions of 8 variables of weight 2. Any pair of functions from the expansion of f0 satisfies the condition: g ⋅ h = 0. The minimal CI-functions of 8 variables under consideration were constructed from the known minimal CI-functions of weights 2, 6, 8 and 10 of a small number of variables. The vector of values of the function f0 and its parameters are given below (all vectors of values are shown in hexadecimal notation).
| wt(f0) = 128 | cor(f0) = 4 | deg(f0) = 3 | nl(f0) = 64 |
One can also obtain the function f0 by considering the sum of 15 minimal CI-functions of 8 variables of weight 8, of one minimal CI-function of 8 variables of weight 6, and of one minimal CI-function of 8 variables of weight 2.
Below we show the 7-stable function f1 of 10 variables. This function is expanded in a sum of 83 minimal CI-functions of 10 variables of weight 6, 2 minimal CI-functions of 10 variables of weight 4, and 3 minimal CI-functions of 10 variables of weight 2. Any pair of functions from the expansion of f1 satisfies the condition g ⋅ h = 0. The minimal CI-functions of 10 variables under consideration were constructed from the known minimal CI-functions of weights 2, 4 and 6 of a small number of variables.
| wt(f1) = 512 | cor(f1) = 7 | deg(f1) = 2 | nl(f1) = 256 |
This function attains its upper nonlinearity bound for 7-stable functions (see, for example, [4]). Note that functions with such parameters may also be constructed by means of the method of [4].
A similar approach was invoked to construct 4- and 6-stable functions which may be expanded in a sum of minimal CI-functions of 11 variables of weight 4, among which any pair of functions satisfies the condition g ⋅ h = 0. Below we give the vectors of values and parameters of the two stable functions functions f2 and f3 of 11 variables.
| wt(f2) = 1024 | cor(f2) = 4 | deg(f2) = 6 | nl(f2) = 768 |
| wt(f3) = 1024 | cor(f3) = 6 | deg(f3) = 4 | nl(f3) = 768 |
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Articles in the same Issue
- On a method of synthesis of correlation-immune Boolean functions
- On the Δ-equivalence of Boolean functions
- On diagnostic tests of contact break for contact circuits
- On stabilization of an automaton model of migration processes
- Bounds on the discrepancy of linear recurring sequences over Galois rings
- Asymptotically best method for synthesis of Boolean recursive circuits
Articles in the same Issue
- On a method of synthesis of correlation-immune Boolean functions
- On the Δ-equivalence of Boolean functions
- On diagnostic tests of contact break for contact circuits
- On stabilization of an automaton model of migration processes
- Bounds on the discrepancy of linear recurring sequences over Galois rings
- Asymptotically best method for synthesis of Boolean recursive circuits
