Number Theoretic Transforms for Secure Signal Processing

Abstract

Multimedia contents are inherently sensitive signals that must be protected whenever they are outsourced to an untrusted environment. This problem becomes a challenge when the untrusted environment must perform some processing on the sensitive signals; a paradigmatic example is Cloud-based signal processing services. Approaches based on Secure Signal Processing (SSP) address this challenge by proposing novel mechanisms for signal processing in the encrypted domain and interactive secure protocols to achieve the goal of protecting signals without disclosing the sensitive information they convey.This work presents a novel and comprehensive set of approaches and primitives to efficiently process signals in an encrypted form, by using Number Theoretic Transforms (NTTs) in innovative ways. This usage of NTTs paired with appropriate signal pre-and post-coding enables a whole range of easily composable signal processing operations comprising, among others, filtering, generalized convolutions, matrix-based processing or error correcting codes. Our main focus is on unattended processing, in which no interaction from the client is needed; for implementation purposes, efficient lattice-based somewhat homomorphic cryptosystems are used. We exemplify these approaches and evaluate their performance and accuracy, proving that the proposed framework opens up a wide variety of new applications for secured outsourced-processing of multimedia contents.

Efficient PRNU Matching in the Encrypted Domain

“…In [5], we proposed a fully unattended solution, based on RLWE (Ring Learning with Errors) cryptosystems [6], which is able to homomorphically evaluate the PRNU extraction/detection without the need of the intervention of the secret key owner in the middle of the process. To this aim, we resort to the pre-/post processing introduced in [7] and the homomorphic Wavelet-denoising proposed in [8].…”

Revisiting Multivariate Ring Learning with Errors and Its Applications on Lattice-Based Cryptography

“…Precisely, as in many real scenarios the number of required operations is known beforehand, SHE turns out to be a perfect fit. Furthermore, many optimizations have been incorporated and, consequently, lattice-based SHE/FHE cryptosystems are being progressively adopted by researchers in the field [7,[29][30][31]. In particular, RLWE-based cryptosystems show nowadays the best runtime performance.…”

“…Additionally, whenever we reduce modulo q and work over R q , we can impose (for the sake of efficiency) that both polynomials functions x 64 + 1 and y 27 + 5 factor in linear terms enabling the use of variants of the NTT. Additionally, slot encoding and slot manipulations are still possible in the plaintext ring by means of the pre-/post-processing, as presented in [7]. Analogously to the negayclic convolution, these pre-/post-processing steps preserve the properties of the NTT transform over a ring with an α-generalized convolution [36].…”

“…Among the possible quotient polynomial rings used for this purpose, the most practical ones are those where the ideal is a cyclotomic polynomial of the form 1 + z n , with n a power of two. They present two advantages: (a) they enable efficient implementations of polynomial operations through fast radix algorithms of the Number Theoretic Transforms (NTTs) [5,6], and (b) the polynomial operations over the used ring correspond to basic blocks in practical applications in Com-puter Vision and Signal Processing [7][8][9], comprising, among others, linear convolutions, filtering, and linear transforms.…”

Performance analysis of number theoretic transform-based convolution using field programmable gate array

“…The NTT can be effectively used in the lossless medical image transmission by water marking approach [2]. The NTT's effectiveness has been proven in the images lossless transmission and in convolution fast calculation [3] The NTT combined with lattice based cryptosystems performs cyclic, negacyclic and convolutions in encrypted domain [4].NTT algorithm are improvised for the float value transforms for Fourier, Hartley, Sine and Cosine signals [5] This paper proposes the design of convolution based on NTT algorithm using the VHDL coding. The VHDL coding of the FFT based convolution is time consuming in design and occupies more number of multipliers and adders.…”