The Chinese remainder theorem (CRT) makes it possible to reduce modular arithmetic calculations with large moduli to similar calculations for each of the factors of the modulus. At the end, the outcomes of the subcalculations need to be pasted together to obtain the final answer. The big advantage is immediate: almost all these calculations involve much smaller numbers.
For instance, the multiplication \(24 \times 32 \pmod{35}\) can be found from the same multiplication modulo 5 and modulo 7, since \(5 \times 7=35\) and these numbers have no factor in common. So, the first step is to calculate:
The CRT, explained for this example, is based on a unique correspondence (Figure 1) between the integers \(0, 1, \ldots, 34\) and the pairs \((u,v)\) with \(0 \leq u <5\) and \(0 \leq v<7.\) The mapping from \(i, 0 \leq i < 35,\) to the pair \((u,v)\) is given by the reduction of i modulo 5 and modulo 7, so \(i=24\) is mapped to \((u,v)=(4,3).\) The mapping from \((u,v)\) back to i is given by \(...
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References
Shapiro, H.N. (1983). Introduction to the Theory of Numbers. John Wiley & Sons, New York.
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van Tilborg, H. (2005). Chinese Remainder Theorem. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_58
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