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Concrete Multiplicative Complexity of Symmetric Functions



Joan Boyar, Rene Peralta


The multiplicative complexity of a Boolean function f is defined as the minimum number of binary conjunction (AND) gates required to construct a circuit representing f, when only exclusive-or, conjunction and negation gates may be used. This article explores in detail the multiplicative complexity of symmetric Boolean functions. New techniques that allow such exploration are introduced. They are powerful enough to give exact multiplicative complexities for several classes of symmetric functions. In particular, the multiplicative complexity of computing the Hamming weight of n bits is shown to be exactly n-Hnum(n), where Hnum(n) is the Hamming weight of the binary representation of n. We also show an intriguing relationship between the complexities of symmetric functions and fractals derived from the parities of binomial coefficients.
Conference Location
, 1
Conference Title
MFCS 2006


circuit complexity, cryptographic proofs, multiplicative complexity, symmetric functions


Boyar, J. and Peralta, R. (2006), Concrete Multiplicative Complexity of Symmetric Functions, MFCS 2006, , 1, [online], (Accessed July 19, 2024)


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Created July 31, 2006, Updated October 12, 2021