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**2022/1077**

**Title:**

New Bounds on the Multiplicative Complexity of Boolean Functions

**Authors:**
Meltem Sonmez Turan

**Abstract:**

Multiplicative Complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis AND, XOR, NOT. This complexity measure is relevant for many advanced cryptographic protocols such as fully homomorphic encryption, multi-party computation, and zero-knowledge proofs, where processing AND gates is more expensive than processing XOR gates. Although there is no known asymptotically efficient technique to compute the MC of a random Boolean function, bounds on the MC of Boolean functions are successfully used to to show existence of Boolean functions with a particular MC.

In 2000, Boyar et al. showed that, for all n\geq 0, at most 2^{k^2+2k+2kn+n+1} n-variable Boolean functions can be computed with k AND gates. This bound is used to prove the existence of a 8-variable Boolean functions with MC greater than 7. In this paper, we improve the Boyar et al. bound.

**ePrint:**
https://eprint.iacr.org/2022/1077

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