Groups assignment

Group Theory

Due Thursday November 15: 11:00 A.M.

Symmetries of Square		Consider the symmetries of the square group introduced here. 1. Compute the operations given on p. 16. in Exercise 1. 2. Verify associativity for one set of 3 elements of the group as described in point 4 on p. 18. 3. Find the square roots of I as described in point 5 on p. 18.
Euler's Code		Consider Euler's Code, introduced in Section 4.4.3 of this. 1. Using 17 as the value for p (the prime modulus), encrypt and decrypt the letter j using Euler's Code. Use 3 or 4 for your encryption key, whichever is better. Defend your choice. Show the calculations determining your decryption key and verify that it is an inverse of your encryption key in the right modulus. 2. The inverse of the encryption key is computed two different ways in the example worked out for you in Section 4.4.3. You may use either method (using Euler's theorem is probably simpler than using Euclid's algorithm). But whichever way you use, show your work.

Symmetries
of Square

Consider the symmetries of the square group introduced here.

Compute the operations given on p. 16. in Exercise 1.

Verify associativity for one set of 3 elements of the group as described in point 4 on p. 18.

Find the square roots of I as described in point 5 on p. 18.

Euler's Code

Consider Euler's Code, introduced in Section 4.4.3 of this.

Using 17 as the value for p (the prime modulus), encrypt and decrypt the letter j using Euler's Code. Use 3 or 4 for your encryption key, whichever is better. Defend your choice. Show the calculations determining your decryption key and verify that it is an inverse of your encryption key in the right modulus.

The inverse of the encryption key is computed two different ways in the example worked out for you in Section 4.4.3. You may use either method (using Euler's theorem is probably simpler than using Euclid's algorithm). But whichever way you use, show your work.