On Solving a Generalized Chinese Remainder Theorem in the Presence of Remainder Errors

In estimating frequencies given that the signal waveforms are undersampled multiple times, Xia et. al. proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are $M_1, M_2, \cdots, M_k$ which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within \tau=\sds \max_{1\le i\le k} \min_{\stackrel{1\le j\le k}{j\neq i}} \frac{\gcd(M_i,M_j)}4, their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than $\tau$. Accurately finding the quotients is a critical ingredient in their approach. In this paper, we shall start with a faithful historical account of the generalized CRT. We then present two treatments of the problem of solving generalized CRT with erroneous remainders. The first treatment follows the route of Wang and Xia to find the quotients, but with a simplified process. The second treatment considers a simplified model of generalized CRT and takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo $4\tau$. Both of our treatments produce efficient algorithms with essentially optimal performance. Finally, this paper constructs a counterexample to prove the sharpness of the error bound $\tau$

A Randomized Sublinear Time Parallel GCD Algorithm for the EREW PRAM

We present a randomized parallel algorithm that computes the greatest common divisor of two integers of n bits in length with probability 1-o(1) that takes O(n loglog n / log n) expected time using n^{6+\epsilon} processors on the EREW PRAM parallel model of computation. We believe this to be the first randomized sublinear time algorithm on the EREW PRAM for this problem

On the complexity of solving linear congruences and computing nullspaces modulo a constant

We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k>1. These problems are known to be complete for the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case that k is prime (Buntrock et al, 1992). By considering variants of standard logspace function classes --- related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k --- we show that these problems of linear algebra are also complete for {coMod_k L} for any constant k>1. Our results are obtained by defining a class of functions FUL_k which are low for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1 reductions (including {Mod_k L} oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL_k for arbitrary moduli k to the class {F.coMod_k L} of functions whose output symbols are verifiable by {coMod_k L} algorithms; and consider what consequences such a comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L} = {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to presentation, new observations regarding the prospect of oracle closures. Comments welcom

Computing Hilbert class polynomials with the Chinese Remainder Theorem

We present a space-efficient algorithm to compute the Hilbert class polynomial H_D(X) modulo a positive integer P, based on an explicit form of the Chinese Remainder Theorem. Under the Generalized Riemann Hypothesis, the algorithm uses O(|D|^(1/2+o(1))log P) space and has an expected running time of O(|D|^(1+o(1)). We describe practical optimizations that allow us to handle larger discriminants than other methods, with |D| as large as 10^13 and h(D) up to 10^6. We apply these results to construct pairing-friendly elliptic curves of prime order, using the CM method.Comment: 37 pages, corrected a typo that misstated the heuristic complexit