A Robust Generalized Chinese Remainder Theorem for Two Integers

Abstract

A generalized Chinese remainder theorem (CRT) for multiple integers from residue sets has been studied recently, where the correspondence between the remainders and the integers in each residue set modulo several moduli is not known. A robust CRT has also been proposed lately for robustly reconstruct a single integer from its erroneous remainders. In this paper, we consider the reconstruction problem of two integers from their residue sets, where the remainders are not only out of order but also may have errors. We prove that two integers can be robustly reconstructed if their remainder errors are less than M/8, where M is the greatest common divisor (gcd) of all the moduli. We also propose an efficient reconstruction algorithm. Finally, we present some simulations to verify the efficiency of the proposed algorithm. The study is motivated and has applications in the determination of multiple frequencies from multiple undersampled waveforms. Index TermsChinese remainder theorem (CRT), robust CRT, dynamic range, residue sets, remainder errors, frequency determination from undersampled waveforms

Multi-Channel Modulo Samplers Constructed From Gaussian Integers

“…We now consider the reconstruction of g k from noisy modulo samples. Since CRT is applicable to Gaussian integers, one can use any existing recovery algorithms by replacing positive integers with Gaussian ones [14], [19]- [28]. We consider robust CRT (RCRT) [14] as it has a closed-form solution with fast implementation.…”

A Secret Image Restoring Scheme Using Threshold Pairs of Unordered Image Shares

“…Without considering the correspondence between a share's pixel and an original image, the problem is to determine {p i , q i } from their pixel pairs 3 } of the three shares with moduli m 1 , m 2 , and m 3 , respectively. This can be solved by the generalized CRT, which was first studied in [22] and later developed in [23], [24]. Inspired by these works, in this paper, we apply the generalized CRT to share and recover two images.…”

“…For the recovery algorithm, the problem is modeled as a generalized CRT, where two secret images are reconstructed simultaneously from their unordered shares. This type of generalized CRT was first studied in [22] and later developed independently in [23], [24]. Compared with existing schemes, the proposed sharing algorithm is more secure, and the recovery algorithm is more effective.…”

Unordered Multisecret Sharing Based on Generalized Chinese Remainder Theorem

“…is paper gives a novel scheme to share and recover multisecret from the unordered shares. Motivated by the works in [25][26][27], we propose a generalized CRT-based multisecret-sharing and recovering scheme. e proposed scheme is not a perfect SS since information can be leaked.…”

“…Let p � 10. By Step 3,in Table 4, we select two secrets s 1 , s 2 � 5, 9 { }, which are satisfied with the condition in (27). By Step 4, we select α � 55 and then obtain x 1 , x 2 � 555, 559 { }.…”