## Exercises7.8Sage Exercises

### 1.

Construct a keypair for Alice using the first two primes greater than $$10^{12}\text{.}$$ For your choice of $$E\text{,}$$ use a single prime number and use the smallest possible choice.

Output the values of $$n\text{,}$$ $$E\text{,}$$ and $$D$$ for Alice. Then use Sage commands to verify that Alice's encryption and decryption keys are multiplicative inverses.

### 2.

Construct a keypair for Bob using the first two primes greater than $$2\cdot 10^{12}\text{.}$$ For your choice of $$E\text{,}$$ use a single prime number and use the smallest possible choice. Output the values of $$n\text{,}$$ $$E\text{,}$$ and $$D$$ for Alice.

Encode the word Math using ASCII values in the same manner as described in this section (keep the capitalization as shown). Create a signed message of this word for communication from Alice to Bob. Output the three integers: the message, the signed message and the signed, encrypted message.

### 3.

Demonstrate how Bob converts the message received from Alice back into the word Math. Output the value of the intermediate computations and the final human-readable message.

### 4.

Create a new signed message from Alice to Bob. Simulate the message being tampered with by adding $$1$$ to the integer Bob receives, before he decrypts it. What result does Bob get for the letters of the message when he decrypts and unsigns the tampered message?

### 5.Classroom Exercise.

Organize a class into several small groups. Have each group construct key pairs with some minimum size (digits in $$n$$). Each group should keep their private key to themselves, but make their public key available to everybody in the room. It could be written on the board (error-prone) or maybe pasted in a public site like pastebin.com 9 . Then each group can send a signed message to another group, where the groups could be arranged logically in a circular fashion for this purpose. Of course, messages should be posted publicly as well. Expect a success rate somewhere between 50% and 100%.

If you do not do this in class, grab a study buddy and send each other messages in the same manner. Expect a success rate of 0%, 50% or 100%.

pastebin.com