Math 1372 Spring Semester 1999 2000

Math 5248 Lecture 2 Spring Semester 2013 Cryptology and Number Theory

Instructor: David Frank, 352 Vincent Hall, 612-625-7396, [email protected]

Office Hours (subject to change): Tuesdays and Thursdays 1:00-2:30.

Text: Cryptology and Number Theory, notes by Paul Garrett, available at Alpha Print in Dinkytown,

Attendance It is to your benefit to attend class regularly. HOWEVER, I am on medication which compromises my immune system. Therefore, IF YOU ARE ILL, STAY HOME.

Prerequisite: Two semesters of sophomore math

Exams: Midterms will be given during usual class times: Midterm I Feb 19 Midterm II, April 9. The Final Exam, in accordance with the official class schedule, will be given 8:00 a.m.–10:00 a.m., Friday, May 17, in a room TBA. Please do not ask to take the final exam at an earlier date. Although the last date of instruction is Friday, May 10, the last day of final examinations and the end of spring semester is May 18. Make your plans accordingly.

Grading Policy Midterm I 250 pts Midterm II 250 pts Homeworks 100 pts Final Exam 400 pts TOTAL 1000 pts

Homework The assigned problems are due at the beginning of lecture.

I encourage you to discuss homework problems with your classmates, including strategies for solving different kinds of problems. Indeed, this is one of the best ways to improve your understanding of the course. However, when you actually write up your solutions, you must do this on your own. Handing in homework which is almost identical to a classmate's is a form of cheating.

Your homework should be:  neatly and clearly written, using complete sentences and correct grammar. You must show your method of solution as well as your answer. Neat crossing out is acceptable, but homework which looks like scrap paper, which contains only answers and no method of solution, or which appears to have been written at the last minute or without reasonable care, will not receive full credit and may not be graded.  handed in on time, at the beginning of class. Late homework will be not be graded,  stapled (not dog-eared), in order to form a more perfect union and promote the general welfare. If your homework is not stapled, there is a high probability that only the first page will be returned and the other pages will be lost.

Make-ups Students must make arrangements in advance if they will not be handing in homework on time or will miss an exam. Exam absences due to recognized University related activities, religious holidays, verifiable illness, and family/medical emergencies will be dealt with on an individual basis.

SCHOLASTIC CONDUCT: I expect the highest standards of conduct from members of this class. Cases of academic dishonesty will be treated with utmost seriousness. More generally, I expect students to be actively involved with the issues presented in class. In particular, you should be prepared each week to ask and answer questions.

INCOMPLETES: I will consider giving you an incomplete if you have successfully completed all but a small portion of the work of the course and some severe, unexpected event prevents you from completing the course. This means that you must have taken both midterms and must be doing at least C- work. I cannot give you an incomplete simply because you are behind in your work. In that case, you should try to drop the course.

S/N GRADE: If you are registered S/N, I will submit a grade of S if your letter grade is C- or above; I will submit a grade of N if your letter grade is D+ or below.

Baby vs. Real Problems: Often a real-world problem might involve an integer n having 100 digits, and might need a fast computer to solve. On a homework assignment or an exam, I might give you a problem using similar concepts and solved using similar steps, but with n being an integer less than 100. Your calculator may well have a command which will solve this baby problem almost instantly. Therefore, even though you will be allowed to use calculators on exams and homework, you must show all your steps to get any credit.

Schedule

1/22/2013 Ch 1 1/24/2013 Ch 2 1/29/2013 Ch 2 1/31/2013 Ch 4 2/5/2013 Ch 7 (RSA, Diffie-Hellman, ElGamel)+HW1

Having introduces these public-key ciphers, we will now study the underlying mathematical issues: (a) What is φ(x) and why is xφ(n)≡1 mod n (at least when gcd(x,n)=1)? (b) How do we find large primes? (c) What kinds of attacks might factor large composite numbers? (d) How do we do fast exponentiation; that is, how do we calculate xk≡n, when k is large? (e) For ElGamel, we need to study issues relating to the equation bx≡c mod n; in particular, the existence of primitive roots

2/7/2013 Fast exponentiation (9.5), Density of primes (8.2) Euler phi-function (10.6) 2/12/2013 Discussion of (Z/n)x; Proof of Euler’s Theorem (10.6) 2/14/2013 10.1-10.3+HW2 2/19/2013 Midterm 1 2/21/2013 10.1-10.3, 10.5 2/26/2013 13,1, 13.2, 13.5, 13.6 2/28/2013 13,1, 13.2, 13.5, 13.6 3/5/2013 18.1-18.5 3/7/2013 18.1-18.5 +HW3 3/12/2013 9.1, 9.5, 9.6, 9.7, 14.4 3/14/2013 10.8, Ch. 23 3/19/2013 Break 3/21/2013 Break 3/26/2013 Ch. 23 3/28/2013 Ch. 23 4/2/2013 Ch 17+HW4 4/4/2013 review 4/9/2013 Midterm 2 4/11/2013 17.2, 17.3 4/16/2013 Ch. 17 4/18/2013 Ch. 16 4/23/2013 Ch. 16 4/25/2013 Ch. 12 4/30/2013 Ch. 12+Review 5/2/2013 HW5+Review 5/7/2013 Review 5/9/2013 Review

5/17/2013 Final Exam 8:00AM-10:00AM. The location of the final exam is VINCENT HALL 16.