Chapter 7

What is the total number of possible monoalphabetic cryptosystems? How secure are such cryptosystems?

Prove that a \(2 \times 2\) matrix \(A\) with entries in \(\mathbb{Z}_{26}\) is invertible if and only if \(\operatorname{gcd}(\operatorname{det}(A), 26)=\) 1

Encrypt each of the following RSA messages \(x\) so that \(x\) is divided into blocks of integers of length 2 ; that is, if \(x=142528\), encode 14,25 , and 28 separately. (a) \(n=3551, E=629, x=31\) (b) \(n=2257, E=47, x=23\) (c) \(n=120979, E=13251, x=142371\) (d) \(n=45629, E=781, x=231561\)

Decrypt each of the following RSA messages \(y\). (a) \(n=3551, D=1997, y=2791\) (b) \(n=5893, D=81, y=34\) (c) \(n=120979, D=27331, y=112135\) (d) \(n=79403, D=671, y=129381\)

For each of the following encryption keys \((n, E)\) in the RSA cryptosystem, compute \(D .\) (a) \((n, E)=(451,231)\) (b) \((n, E)=(3053,1921)\) (c) \((n, E)=(37986733,12371)\) (d) \((n, E)=(16394854313,34578451)\)

Find integers \(n, E,\) and \(X\) such that $$X^{E} \equiv X \quad(\bmod n)$$ Is this a potential problem in the RSA cryptosystem?

Every person in the class should construct an RSA cryptosystem using primes that are 10 to 15 digits long. Hand in \((n, E)\) and an encoded message. Keep \(D\) secret. See if you can break one another's codes.