Faster Homomorphic Evaluation of Discrete Fourier Transforms

Abstract

Abstract. We present a methodology to achieve low latency homomorphic operations on approximations to complex numbers, by encoding a complex number as an evaluation of a polynomial at a root of unity. We then use this encoding to evaluate a Discrete Fourier Transform (DFT) on data which has been encrypted using a Somewhat Homomorphic Encryption (SHE) scheme, with up to three orders of magnitude improvement in latency over previous methods. We are also able to deal with much larger input sizes than previous methods. Due to the fact that the entire DFT algorithm is an algebraic operation over the underlying ring of the SHE scheme (for a suitably chosen ring), our method for the DFT utilizes exact arithmetic over the complex numbers, as opposed to approximations.

Homomorphic SIM$$^2$$D Operations: Single Instruction Much More Data

“…Example 2. In the case of the 2-power cyclotomic polynomial f pXq " X 2 k`1 , the above map replaces negative powers X´j by´X d´j , which coincides with the approach from [14]: when expressed in terms of the basis 1, X, X 2 , . .…”

“…, b´1u is used in practice, such as binary b " 2 or ternary b " 3. For use in SHE schemes, several variations [17,14,12,3] have been proposed. For the purposes of this paper we mention the non-integral base non-adjacent form (NIBNAF) from [3] which is a very sparse expansion w.r.t.…”

“…has been represented as a Laurent polynomial apXq P ZrX˘1s. Encoding such a Laurent polynomial in the plaintext space R t has been considered in a series of recent works [17,14,12,3]. However, it was emphasized in [14] that the plaintext space should only be defined modulo a 2-power cyclotomic polynomial f pXq " X 2 k`1 for some k. The reason for this restriction is that the authors required a small and sparse representation for X´1, which in this case is given by X´1 "´X 2 k´1 mod f pXq.…”

“…The noise growth is influenced by d, σ, but also by the plaintext modulus t. A first optimization to decrease noise growth is therefore to use a smaller plaintext space. Several encoding techniques [23,17,14,12,3,10] have been proposed whose goal is to 'spread out' the numerical input data as evenly as possible over the whole plaintext space, resulting in smaller t. A second optimization, which can be combined with the first, is to decompose the plaintext space into smaller pieces using the Chinese Remainder Theorem (CRT) and run several computations in parallel [25,4]. Smart and Vercauteren [25] described how to carry out SIMD calculations in a SHE context by viewing R t as the CRT composition of Z t rXs{pf 1 pXqqˆZ t rXs{pf 2 pXqqˆ¨¨¨ˆZ t rXs{pf r pXqq, where f 1 pXqf 2 pXq¨¨¨f r pXq is a factorization of f pXq into coprime factors.…”

Efficiently Processing Complex-Valued Data in Homomorphic Encryption

“…Since the publication of Gentry's seminal work [15], this research area has evolved rapidly and is on the verge of reaching a first degree of maturity, as was recently demonstrated e.g. by practical implementations of privacy-enhanced electricity load forecasting [2,4], digital image processing [1,10], and medical data management [8,12,17]. Most of the current focus lies on somewhat homomorphic encryption (SHE), where the schemes are capable of homomorphically evaluating an arithmetic circuit having a certain predetermined computational depth.…”