Cipher 7: Generalised RSA

The RSA cryptosystem is named after Rivest, Shamir and Adleman, who rediscoved it at MIT. It was created earlier by Clifford Cocks at GCHQ, but that information was classified. It is an example of a trapdoor cipher (others include elliptic curve cryptography), where different keys are required for encoding and decoding information.

Specifically, one encrypts some plaintext x into ciphertext y by using the formula y = x^e (mod N), where e and N are parts of the public key called the encoding exponent and modulus, respectively. By Euler’s extension of Fermat’s little theorem, it is possible to reverse this process if you have a number d (the decoding exponent); specifically, x = y^d (mod N). It is difficult to obtain d from e unless you know the prime factorisation of N.

I’m going to tell you that in this case the encoding exponent e = 65537 and the modulus N = 2^1024+1. The ciphertext y is the 308-digit integer given below:


Once you recover x, you’ll have to find some way of converting the integer into a message. The password for the secret area, by the way, is entirely in lower-case.

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