On Some Algebraic Properties of the Chinese Remainder Theorem with Applications to Real Life (original) (raw)

The General Chinese Remainder Theorem

The American Mathematical Monthly, 1952

The Chinese remainder theorem deals with systems of modular equations. The classical variant requires the modules to be pairwise coprime. In this paper we discuss the general variant, which does not require this restriction on modules. We have selected and implemented several algorithms for the general Chinese remainder theorem. Moreover, we point out some interesting applications of this variant in secret sharing and threshold cryptography.

Further results on Chinese remaindering

1997

We present an attack on the RSA cryptosystem in a case where the attacker has very few knowledge of the parameters: even the public modulus is unknown to him. Our attack is based on the presence of faults and on the use of the Chinese Remainder Theorem to perform computations. With this attack, a damaged smart card can partially be broken.

A generalization of the Chinese remainder theorem

2001

The Chinese Remainder Theorem is more than 2000 years old. About this, e can read in the [1]. The Theorem has been successfully applied in the algorithm recently developed for the calculus with the large numbers, and several elementary arithmetics problems are built onto this.

Multivariable Chinese remainder theorem

Resonance, 2015

In this note we show a multivariable version of the Chinese remainder theorem: a system of linear modular equations ai1xi + ... + ainxn = bi mod mi, i = 1, ..., n has solutions if mi > 1 are pairwise relatively prime and in each row, at least one matrix element aij is relatively prime to mi. The solution x can be found in a parallelepiped of volume M = m1m2 • • • mn. The Chinese remainder theorem is the special case, where A has only one column and the parallelepiped has dimension 1 × 1 × ... × 1 × M .

The Chinese Remainder Theorem, Associative Algebras, and Multiplicative Complexity

1990 Conference Record Twenty-Fourth Asilomar Conference on Signals, Systems and Computers, 1990.

Polynomial multiplication continues to play a fundamental role in many important algorithms (e.g., convolution, correlation). Many methods have been developed which facilitate highly efficient polynomial multiplication. One such method is baaed on the Chinese Remainder Theorem (CRT), a classic result from ring theory. The CRT is known to reduce the complexity of polynomial multiplication from O (N 2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algebras. This new point of view provides a clearer understanding o€ the CRT.

The complexity of the Chinese Remainder Theorem

arXiv (Cornell University), 2023

The Chinese Remainder Theorem for the integers says that every system of congruence equations is solvable as long as the system satisfies an obvious necessary condition. This statement can be generalized in a natural way to arbitrary algebraic structures using the language of Universal Algebra. In this context, an algebra is a structure of a first-order language with no relation symbols, and a congruence on an algebra is an equivalence relation on its base set compatible with its fundamental operations. A tuple of congruences of an algebra is called a Chinese Remainder tuple if every system involving them is solvable. In this article we study the complexity of deciding whether a tuple of congruences of a finite algebra is a Chinese Remainder tuple. This problem, which we denote CRT, is easily seen to lie in coNP. We prove that it is actually coNP-complete and also show that it is tractable when restricted to several well-known classes of algebras, such as vector spaces and distributive lattices. The polynomial algorithms we exhibit are made possible by purely algebraic characterizations of Chinese Remainder tuples for algebras in these classes, which constitute interesting results in their own right. Among these, an elegant characterization of Chinese Remainder tuples of finite distributive lattices stands out. Finally, we address the restriction of CRT to an arbitrary equational class V generated by a two-element algebra. Here we establish an (almost) dichotomy by showing that, unless V is the class of semilattices, the problem is either coNP-complete or tractable.

Three Novel Theorems for Applied Cryptography

International Journal of Computer Applications, 2013

With advancements in computing capabilities public key cryptosystems are going to be more complex yet vulnerable over the modern day"s computer networks and associated security mechanism, especially those based on novel approaches of applied mathematics. This paper explores three novel theorems derived while studying and implementing RSA algorithm, one of the strongest public key cryptosystem. The proposed Theorems are best suited and adequate for RSA algorithm yet being applicable to some of other existing algorithms and theorems of applied mathematics. The first theorem deals with concept of ambiguity while calculating multiplicative inverse of encryption key which in some of instances returns undesirable negative numbers not useful as decryption key. Second theorem deals with unconcealed multiplicative inverses, unconcealed are values which remain unchanged after any mathematical transformations. Concept of unconcealed multiplicative inverses is useful in key generation for RSA cryptosystem. Third theorem deals with the concept of unconcealed exponentiation modulo quite useful in finding unconcealed signature and messages to form UM Matrix for RSA.

On the reduction in multiplicative complexity achieved by the polynomial residue number system

IEEE Transactions on Signal Processing, 1992

The polynomial residue number system (PRNS) is known to reduce the complexity of polynomial m iltiplication from O(N2) to O (N). A new interpretation of this complexity reduction is given in the context of associative algt,bras over a finite field. The new point of view provides a clearer understanding of the Chinese remainder theorem.