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Tight Bounds for the 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 − H^N(n), where H^N(n) is the Hamming weight of the binary representation of n. We also show a close relationship between the complexities of basic symmetric functions and the fractal known as Sierpinski's gasket.
Theoretical Computer Science


circuit complexity, cryptographic proofs, multi-party computation, multiplicative complexity, symmetric functions


Boyar, J. and Peralta, R. (2008), Tight Bounds for the Multiplicative Complexity of Symmetric Functions, Theoretical Computer Science, [online],, (Accessed April 23, 2024)
Created April 27, 2008, Updated October 14, 2021