Chinese remainder theorem in cryptography

\( \newcommand\] We can solve such systems of equations using what is called (in the West) the Chinese Remainder Theorem (CRT). Below is one of the simpler formations of the Chinese Remainder Theorem, involving only two equations/moduli (unlike the example above, which has three moduli 3,5 , and 7 ) Theorem \(13.9\)(CRT) Suppose \(\operatorname\). So any two solutions to this system of equations are congruent \(\bmod r s\) CRT Encodings Preserve Structure Let’s call \((u, v) \in \mathbb\) encoding." The amazing thing about these CRT encodings is that they preserve all sorts of arithmetic structure. Example Taking \(r=3\) and \(s=5\), let’s write down the CRT encodings of every element in \(\mathbb\), and I encourage you to try it! As if that weren’t amazing enough, the same thing holds for multiplication: Example Let’s return to the \(r=3, s=5\) setting for \(C R T\) and highlight \(x=6, x^\) integer. The same idea works for multiplication as well, giving us the following: Application to RSA You might be wondering what the point of all of this is. \(\) The CRT method seems like a very indirect and wasteful way to compute anything. This impression might be true for simple operations like addition and single multiplications. However, the CRT method is faster for exponentiation \(\bmod N\), which is the main operation in RSA! Example In Sage, we can do basic exponentiation mod \(n\) as follows: If we are working over an RSA modulus and know its factorization \(p \times q\), t...

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Exercise from P.71 of Modern Cryptography and Elliptic Curves - A Beginner’s Guide. Prove the Chinese Remainder Theorem. We proved the special case with $n = 2$ Assume that we have proved it for some $n \geq 2$. Now, let a system of congruences be given. \[\begin$. Subscribe Share:

Liberato A, Martinello M, Gomes R, Beldachi A, Salas E, Villaca R, Ribeiro M, Kanellos G, Nejabati R, Gorodnik A and Simeonidou D (2018). RDNA: Residue-Defined Networking Architecture Enabling Ultra-Reliable Low-Latency Datacenters, IEEE Transactions on Network and Service Management, 15 :4, (1473-1487), Online publication date: 1-Dec-2018. • Reviewer: James Harold Davenport The focus of this book is definitely on the Chinese remainder theorem (CRT) and the corresponding algorithm. Unusually, but most interestingly, there is an excellent historical introduction to the CRT in both the Chinese and the European mathematical traditions. For example, Fibonacci's description is translated, as are old Chinese applications. (One could regret that no history is given for other parts of mathematics.) The book then covers the history of the theorem, in settings ranging from the very numeric to the ideal-theoretic. Applications in computing follow, with a chapter on the algorithms known as “modular” by the symbolic computation community; these algorithms generally rely on some variant of the theorem to produce their final results. The authors describe the Scho¨nhage modular algorithm, with emphasis on the efficient special case of the Chinese remainder algorithm. The Collins modular resultant, one of the classics of modular computation, is also described. Unfortunately, no indication is given that the Scho¨nhage algorithm has been overtaken. Another chapter describes issues such as po...