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In mathematics, a finitary boolean function is a function of the form f Bk → B, where B = is a boolean domain and where k is a nonnegative integer. In the case where k = 0, the "function" is simply a constant element of B. More generally, a function of the form f X → B, where X is an arbitrary set, is a boolean-valued function. If X = M = , then f is a binary sequence, that is, an infinite sequence of 0's and 1's. If X = ''k'' = , then f is a binary sequence of length k. There are such functions. These play a basic role in questions of complexity theory as well as the design of circuits and chips for digital computers. The properties of boolean functions play a critical role in cryptography, particularly in the design of symmetric key algorithms (see S-box). A boolean mask operation on boolean-valued functions combines values point-wise (for example, by XOR, or other boolean operators).
Algebraic Normal Form A boolean function can be written uniquely as a sum (XOR) of products (AND). This is known as the Algebraic Normal Form (ANF). where . The values of the sequence can therefore also uniquely represent a boolean function. The algebraic degree of a boolean function is defined as the highest number of that appear in a product term. Thus has degree 1 (linear), whereas has degree 3 (cubic). See also | ||||||||
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