| It is well known that the Chinese Remainder Theorem(CRT)is not robust,i.e.,a small error in any remainder may lead to a large reconstruction error of the congruence system.Robust Chinese Remainder Theorem(RCRT)allows the approximate reconstruction of the solution of congruence equations in the case where remainders have errors,so it has a wide range of applications in coding,wireless sensor networks(WSNs),signal processing,and other fields,and it gives rise to the research of robust reconstruction with erroneous remainders.A large integer less than the least common multiple(lcm)of all moduli can be robustly reconstructed from its erroneous remainders by the current scheme when all errors are small.However,unrestricted errors in many applications may also occur due to failures or interferences.The existing solution for this problem is based on a strict condition,i.e.,the definition of small error is too strict,so the solution has a high value in theoretical research but it is difficult to be used in practical applications.In order to solve the problems mentioned above,this thesis constructs two RCRT under a relaxed condition and corresponding data aggregation schemes with WSNs as the application scenarios.The main contributions are as follows:1.Construction of RCRT based on consistency adjustment under a relaxed condition.The construction first uses the improved consistency adjustment method in a special Residue Number System with Nonpairwise-Prime Moduli(RNS-NPM)to make the adjusted erroneous remainders satisfy consistent,i.e.,the necessary and sufficient condition for Generalized Chinese Remainder Theorem(GCRT)to have a solution is satisfied,and then removes some remainders by pruning operation to ensure that the small errors are the same after adjustment.Finally,an RCRT with the ability to tolerate a certain number of unrestricted errors,an arbitrary number of small errors,and the erased of some remainders is constructed.2.Construction of RCRT based on folding number adjustment under a relaxed condition.The construction extends the existing RCRT that satisfies the relaxed condition but requires all errors to be small,then proposes an RCRT that tolerates a certain number of unrestricted errors,an arbitrary number of small errors,and the erased of some remainders.Compared with the first contribution,this construction has a higher encoding rate,but the time complexity of reconstruction may also be higher.3.Based on two constructed RCRT and existing RCRT with a relaxed condition,this thesis combines the WSNs as practical application scenarios and proposes three data aggregation schemes based on RCRT.The schemes not only satisfy energy saving,reliability,and non-plaintext transmission but also do not require the signals to be sparse or independent encoders with prior functions,and can also obtain a reasonable approximation of the measured physical quantity even if there are measurement errors in the sensed data. |
