We propose how to homomorphically evaluate arbitrary univariate and bivariate integer functions such as division. A prior work proposed by Okada et al. (WISTP'18) uses polynomial evaluations such that the scheme is still compatible with the SIMD operations in BFV and BGV schemes, and is implemented with the input domain ℤ257. However, the scheme of Okada et al. requires the quadratic numbers of plaintext-ciphertext multiplications and ciphertext-ciphertext additions in the input domain size, and although these operations are more lightweight than the ciphertext-ciphertext multiplication, the quadratic complexity makes handling larger inputs quite inefficient. In this work, first we improve the prior work and also propose a new approach that exploits the packing method to handle the larger input domain size instead of enabling the SIMD operation, thus making it possible to work with the larger input domain size, e.g., ℤ215 in a reasonably efficient way. In addition, we show how to slightly extend the input domain size to ℤ216 with a relatively moderate overhead. Further we show another approach to handling the larger input domain size by using two ciphertexts to encrypt one integer plaintext and applying our techniques for uni/bivariate function evaluation. We implement the prior work of Okada et al., our improved version of Okada et al., and our new scheme in PALISADE with the input domain ℤ215, and confirm that the estimated run-times of the prior work and our improved version of the prior work are still about 117 days and 59 days respectively while our new scheme can be computed in 307 seconds.

Orthogonal frequency division multiplexing (OFDM) channel estimation is the key technique used in broadband wireless networks. The Doppler frequency caused by fast mobility environments will cause inter-carrier interference (ICI) and degrade the performance of OFDM systems. Due to the severe ICI, channel estimation becomes a difficult task in higher mobility scenarios. Our aim is to propose a pilot-aided channel estimation method that is robust to high Doppler frequency with low computational complexity and pilot overheads. In this paper, the time duration of each estimate covers multiple consecutive OFDM symbols, named a “window”. A close-form of polynomial channel modeling is derived. The proposed method is initialized to the least squares (LS) estimates of the channels corresponding to the time interval of the pilot symbols within the window. Then, the channel interpolation is performed in the entire window. The results of computer simulations and computation complexity evaluations show that the proposed technique is robust to high Doppler frequency with low computation complexity and low pilot overheads. Compared with the state-of-the-art method and some conventional methods, the new technique proposed here has much lower computational complexity while offering comparable performance.

The objective of this paper is to develop the theoretical foundation to the pilot-assisted channel estimation using delay-time domain windowing for the coherent detection of OFDM signals. The pilot-assisted channel estimation using delay-time domain windowing is jointly used with polynomial interpolation, decision feedback and Wiener filter. A closed-form BER expression is derived. The impacts of the delay-time domain window width, multipath channel decay factor, the maximum Doppler frequency are discussed. The theoretical analysis is confirmed by computer simulation.

Whereas the autocorrelation is frequently used for pitch estimation, the resultant estimates usually suffer from inaccuracy. Instead of upsampling, we can improve the accuracy of the estimates by applying polynomial interpolation to the autocorrelation directly. For that purpose, four kernels, which are interpolating quadratic, quadratic-B spline, cubic-B spline, and cubic convolution kernels respectively, have been compared. Experiments show that the cubic B spline kernel shows the best performance, a little inferior to the computationally intensive upsampling procedure. The quadratic B spline kernel shows also reasonable performance with the merit of the further reduced computational complexities compared with the cubic B spline kernel.