Detecting Affine Equivalence Of Boolean Functions And Circuit Transformation

Affine equivalence of Boolean functions has various applications in computer science and modern cryptography, such as circuit design and S-boxes. Existing methods for detecting affine equivalence of Boolean functions work in some cases but not when the truth table of a Boolean function is sparse. To improve previous methods and overcome this limitation, we propose a method by transforming the Boolean function to a function with the property that its function values at the orthonormal basis are all equal to 1 or 0, which narrows down the search space of affine transformations. Our first algorithm has the advantage of getting a smaller search space than previous methods and is especially useful for sparse functions. Specifically, when the Boolean functions are sparse, the search space can be reduced exponentially in average and experiments show the efficiency of our first algorithm. We then present another algorithm to transform one circuit into its equivalent affine circuit by synthesizing a reversible circuit and inserting it in front of the original circuit. To our knowledge, this is the first work to automatically synthesize an affine equivalent circuit for any given circuit and the first to do this by combining reversible circuit and non-reversible circuit.