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A Boolean function with multiple outputs, : {,} {,} with > is a vectorial or vector-valued Boolean function (an S-box in symmetric cryptography). [ 6 ] There are 2 2 k {\displaystyle 2^{2^{k}}} different Boolean functions with k {\displaystyle k} arguments; equal to the number of different truth tables with 2 k {\displaystyle 2^{k}} entries.
Balanced Boolean functions are used in cryptography, where being balanced is one of "the most important criteria for cryptographically strong Boolean functions". [3] If a function is not balanced, it will have a statistical bias, making it subject to cryptanalysis such as the correlation attack.
Garbled circuit is a cryptographic protocol that enables two-party secure computation in which two mistrusting parties can jointly evaluate a function over their private inputs without the presence of a trusted third party. In the garbled circuit protocol, the function has to be described as a Boolean circuit.
The following formula shows that a 4-ary function is bent when its nonlinearity is 6: = = In the mathematical field of combinatorics, a bent function is a Boolean function that is maximally non-linear; it is as different as possible from the set of all linear and affine functions when measured by Hamming distance between truth tables.
The functions studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics , social choice theory , random graphs , and theoretical computer science, especially in hardness of approximation , property testing , and PAC learning .
Siegenthaler showed that the correlation immunity m of a Boolean function of algebraic degree d of n variables satisfies m + d ≤ n; for a given set of input variables, this means that a high algebraic degree will restrict the maximum possible correlation immunity. Furthermore, if the function is balanced then m + d ≤ n − 1. [1]
In cryptography, a boolean function is said to be complete if the value of each output bit depends on all input bits. This is a desirable property to have in an encryption cipher, so that if one bit of the input is changed, every bit of the output has an average of 50% probability of changing. The easiest way to show why this is good is the ...
The function has the same security properties as any (cryptographically secure) pseudorandom function. Specifically it shall be hard to distinguish the output from true randomness. The function is called an Oblivious Pseudorandom Function, because the second-party is oblivious to the function's output. This party learns no new information from ...