# Math 348:01 - Spring 2012

In class examples were given of Diffie-Helman key exchange, ElGamal ciphers and RSA public key encryption in some cases with very small modulus, and some with larger modulus. The data for the larger examples appears below.

Diffie-Helman key exchange example:

```p=123456789011 (a prime)
g= 2  (a primitive root modulo p)
A sends to B 98669960721 (modulo p)
B sends to A 66675120784 (modulo p)

Find the key that A and B have exchanged.
```

El Gamal cipher example:

```p=123456789011 (a prime)
b=2            (a primitive root modulo p)
c=32890691253  ( b raised to the power l)

Suppose that A makes public the data above (but not l) and B encrypts
a plaintext X by choosing a secret r and sending A

the header     b^r=60577479993 (modulo p)
the ciphertext Xc^r=59778139972 (modulo p)

Find the value of X.

```

RSA public key example:

```A publishes the following information

modulus m=15241578759079409808671

encryption key:    e=3429235932423923

Using this data and the RSA public key cipher B sends to A

ciphertext 11482879856055838066711

Find the message that B encrypted and sent to A.

```