Department of Mathematics

Math 5405. Cryptography, Spring 2013

Syllabus for the course

Review for the First Midterm

Review for the Second Midterm

Review for the Final

Savin's Lecture Notes (our online text). Do not distribute.

Trappe and Washington's Book (available at the bookstore).

Homework Assignments

1. Due Thursday, January 17.
Trappe and Washington, #4, 11, 13, 16, 18, 19, 20, 21, 24, 25 on Pages 104-108.

2. Due Thursday, January 24.
Savin, Page 138 # 1,2, Page 142 # 1,2,3, Page 145 # 1,3, Page 150 # 1,2,3.

3. Due Thursday, January 31.
Savin, Page 150 #4 (and factor these numbers using either the $p-1$ or $p+1$ method),
Page 154 #1, Page 156 #1,2.

4. Due Thursday, February 7.
Savin, Pages 159-60 #1-5.
Trappe-W, Pages 215-16 #4,8,10. Page 217 #4.

5. Due Thursday, February 14.

The next prime year is the year 2017.
5 is a primitive element (mod 2017).
I chose a mystery number x and computed 5^x (mod 2017) and got 1077.
Please find my number in three different ways.

(1) 2017 is not such a big prime, so use baby-step/giant-step to find x.
(2) 2016 has only small prime factors, so use Pohlig-Helmann to find x.
(3) Although it is a pain, use Index Calculus to find discrete logs of 2,3,5,7,11,13 and then find x.


6. Due Thursday, February 28.
Savin, Page 165 #1,2. Page 170 #1,2.`

7. Due Thursday, March 7.
Savin, Page 173 #1,2. Page 175 #1,2.`

8. Due Thursday, March 21.
Savin, Pages 180-181 #1-5.

9. Due Thursday, March 28.
Homework Number Nine

10. Due Thursday, April 18.
Trappe-Washington, Problems 1-3 on Pages 314-315. Also the following two programming problems.

A. Given input of two integers n and m, return either:
(i) The gcd(m,n) if the gcd is different from 1, or:
(ii) The inverse of m (mod n) if the gcd is equal to 1.
Your output should make it clear which of the two has occurred.

B. Given input of three integers: (b,c,n) and a pair of ordered pairs (x1,y1) and (x2,y2)
(i) Determine whether the two points are on the elliptic curve y^2 = x^3 + bx + c (mod n)
(ii) If they are on the elliptic curve, either:
(iia) Add them to get a third point, or else
(iib) Explain that they cannot be added and return the reason why.

11. Due at the Final Exam on May 1, 2013.
Trappe-Washington, Problems 2-4 on Page 446, and:
Programming Assignment.
Input a natural number n.
A. Check n for primality using the Miller-Rabin test.
If it is probably (99% sure) to be prime, return "Government Prime." Otherwise.
B. Factor n using the Lenstra algorithm. Return the factors.

Supplementary Notes

A Quick Number Theory Review (without proofs)

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