![]() It is actually fairly easy to see the correctness of the algorithm. Now, the questions in your comments appear to be asking about the details of this recombination step. This can be done using Fermat’s little theorem because p and q.Then, we recombine the that is, we find a number m such that:īecause of the Chinese Remainder Theorem (and because p and q are relatively prime), we can immediately deduce that: which is exactly what we were trying to compute. Note that the exponents are reduced module p-1 and q-1. This scheme is more efficient than computing because these two modular exponentiations both use a smaller exponent and a smaller modulus. ![]() Therefore, we can compute the message from given ciphertext more effienctly as follow: In RSA scheme using the Chinese remainder theorem, the following values are precomputed and stored as a public key: Hence, the solution of this simultaneous congruences is for all k are integers. There exists an unique modulo solution of the system of simultaneous congruences above: For any sequence of integers, there exists an integer x solving the following system of congruence equations: Suppose are positive integers and coprime in pair.
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