chiku: Efficient Probabilistic Polynomial Approximations Library

Fully Homomorphic Encryption (FHE) is a prime candidate to design privacy-preserving schemes due to its cryptographic security guarantees. Bit-wise FHE (e.g., FHEW, T FHE) provides basic operations in logic gates, thus supporting arbitrary functions presented as boolean circuits. While word-wise FHE (e.g., BFV, CKKS) schemes offer additions and multiplications in the ciphertext (encrypted) domain, complex functions (e.g., Sin, Sigmoid, TanH) must be approximated as polynomials. Existing approximation techniques (e.g., Taylor, Pade, Chebyshev) are deterministic, and this paper presents an Artificial Neural Networks (ANN) based probabilistic polynomial approximation approach using a Perceptron with linear activation in our publicly available Python library chiku. As ANNs are known for their ability to approximate arbitrary functions, our approach can be used to generate a polynomial with desired degree terms. We further provide third and seventh-degree approximations for univariate Sign(x) ∈ {−1,0,1} and Compare(a − b) ∈ {0, 2¹,1} functions in the intervals [−1,1] and [−5,−5]. Finally, we empirically prove that our probabilistic ANN polynomials can improve up to 15% accuracy over deterministic Chebyshev’s.