Problems of the day

1.

Write a Maple procedure OneTimePad which improves on the original OneTimePad by both encoding and decoding: if the key is positive, then OneTimePad encodes the message using that key. If the key is negative, then OneTimePad decodes the message using the (negative of) the key.

2.

Solve the following equations (find a solution x with $0 \leq x < 8$):

• $3x=1 \mathop{\rm mod} 8$
• $3x=4 \mathop{\rm mod} 8$
• $6x=0 \mathop{\rm mod} 14$

3.

Show that if a and p are positive integers with $\mathop{\rm gcd}(a, p) =1$ then the equation $ax+b=y \mathop{\rm mod} p$ always has an integer solution xfor any choice of integers b and y.

4.

The following message was encoded using an affine code, the alphabet consisted of the upper case alphabet, followed by a space, followed by the lower case alphabet. Decode the message.

$$\mbox{QZxpPBMopwZxLpXZLoYpmA BpRLApuBmmBLYpYlZZMY}$$