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How we can solve this equation and get the value of M?

$$8 = M^{13} \mod 33$$

not a computer program, but a mathematical operation.

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2 Answers 2

up vote 6 down vote accepted

Factor the modulus, $n$ which was given as 33, to yield $p = 3$ and $q = 11$.

The totient, $\phi$ is $(p - 1)(q - 1) = 20$.

The public exponent, $e$ is given as $13$.

Now compute the private exponent $d$ as the multiplicative inverse of $e \mod \phi$, so $e^{-1}\mod \phi \equiv 17$.

The cipher text, $c$ is given as 8.

Finally compute the message, $M$ as $c^d \mod n \equiv 2$.

No algorithm has been published for factoring large numbers quickly, so the first step is what makes breaking RSA "computationally infeasible." Fast algorithms are known for all of the other operations.

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Since we're working modulo 33, we only need to look at values of M between 0 and 32. (Since $a^k \mod b \equiv (a \mod b)^k \mod b)$.

If we examine these 33 values, we get that it holds for M = 2.

However, this in fact holds for any $M = 2 + 33k$, where k is an integer, as per the argument outlined in the beginning.

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