The number of boolean functions with multiplicative complexity 2

The number of boolean functions with multiplicative complexity 2

Multiplicative complexity is a complexity measure defined as the minimum number of AND gates required to implement a given primitive by a circuit over the basis (AND, XOR, NOT). Implementations of ciphers with a small number of AND gates are preferred in protocols for fully homomorphic encryption, multi-party computation and zero-knowledge proofs. In 2002, Fischer and Peralta [12] showed that t...

Multiplicative complexity of vector valued Boolean functions

We consider the multiplicative complexity of Boolean functions with multiple bits of output, studying how large a multiplicative complexity is necessary and sufficient to provide a desired nonlinearity. For so-called ΣΠΣ circuits, we show that there is a tight connection between error correcting codes and circuits computing functions with high nonlinearity. Combining this with known coding theo...

Circuit Complexity and Multiplicative Complexity of Boolean Functions

In this note, we use lower bounds on Boolean multiplicative complexity to prove lower bounds on Boolean circuit complexity. We give a very simple proof of a 7n/3− c lower bound on the circuit complexity of a large class of functions representable by high degree polynomials over GF(2). The key idea of the proof is a circuit complexity measure assigning different weights to XOR and AND gates.

A relation between additive and multiplicative complexity of Boolean functions

In the present note we prove an asymptotically tight relation between additive and multiplicative complexity of Boolean functions with respect to implementation by circuits over the basis {⊕,∧, 1}. To start, consider a problem of computation of polynomials over a semiring (K,+,×) by circuits over the arithmetic basis {+,×} ∪K. It’s a common knowledge that a polynomial of n variables with nonsca...

The complexity of Boolean functions