DORIAN GOLDFELD
Professor
Mathematics Building, Room 422
Columbia University
New York, NY 10027
Telephone: (212) 854-4304
Fax: (212) 854-8962
e-mail: goldfeld@columbia.edu
Research Interests: Number Theory
Recent Preprints: (pdf files)
Voronoi Formulas on GL(n) (with X. Li) new
Second Moments of GL(2) L-Functions (with A. Diaconu) new
Counting Congruence Subgroups (with A. Lubotzky and L. Pyber)
Multiple Dirichlet Series and Moments of Zeta and L-Functions
(with A. Diaconu and J. Hoffstein)
Modular Forms, Elliptic Curves, and the ABC Conjecture
The Gauss Class Number problem for Imaginary Quadratic Fields
The Elementary proof of the Prime Number Theorem, An Historical Perspective
Links
L-functions and Automorphic forms (Goldfeld Fest) (conference photos)
Automorphic Forms and L-Functions for the Group GL(n,R) (Just published!)
Joint COLUMBIA-CUNY-NYU Number Theory Seminar
Braid Group Cryptography
Bretton Woods Workshop on Multiple Dirichlet Series
Decision Regarding World Record Musky Challenge
Very accurate clock
Teaching, Spring 2008:
MATH V3025Y Making, Breaking Codes 3 pts.
(Tuesday/Thursday 2:40-3:55, Room 312 Math Building)
We will use the book: Introduction to Cryptography with Coding Theory (Second Edition), by Wade Trappe and Lawrence C. Washington, Pearson-Prentice Hall.
Grading: Homework 25% of grade, Midterm 35% of grade, Final 40% of grade.
TA's Min Lee < enoia@math.columbia.edu > , Atanas Atanasov < ava2102@columbia.edu>
Columbia HelpRoom
Midterm: Thursday, March 13, 2008, 2:40pm - 3:55pm in Room 312 Math Building. Midterm-solutions
Final Exam: TBA
Homework: (All homeworks are from the book: Introduction to Cryptography with Coding Theory)
#1 (due in class on Thursday, Jan. 31) page 55, problems: 1, 5, 9, 10, 11, 15, 17, 21, 25a
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#2 (due in class on Thursday, Feb. 7) page 104, problems: 1, 4, 12, 13, 18, 33, 34
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#3(due in class on Thursday, Feb 14) Explicitly construct a finite field of 9 elements. Express the multiplicative group G_26 = {1,3,5,7,9,11,15,17,19,21,23,25} as a union of left cosets of the subgroup H = {1,3,9}.
Page 192, problems 1, 2, 5, 11, 12, 16, 17.
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#4(due in class on Thursday, Feb 21) Page 195, problems 23, 24, Page 215, problems 2, 3, 4, 10, 11.
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#5(due in class on Thursday, Feb 28) Page 239, problems 1, 2(b) Page 252, problems 1, 2, 4, 5, 6.
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#6(due in class on Thursday, March 6) Page 370, problems 1, 2, 3, 4, 5, 6, 7, 12.
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#7(due in class on Thursday, March 13) Page 370, problems 10, 12(a), 16. Consider the elliptic curve version of the Diffie-Hellman key exchange using the curve E: y^2 = x^3 + 2x + 4 (mod 11). Alice's public key consists of the two points G = (0,2), G_a = (7,3) on the curve E, and her secret key is a = 6 where G_a = 6G. Bob's public key consists of the two points G = (0,2), G_b = (6,1) on the curve E, and his secret key is b = 4 where G_b = bG. Show how Alice and Bob can compute the shared secret = abG. What is abG?
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#8(due in class on Thursday, April 3) Page 286, problems 1, 2, 3, 4, Page 294, problems 1, 2, 3, Page 315, problem 3.
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#9(due in class on Thursday, April 10) Page 108, problems 25, 26, 27, Page 322, problems 2, 3.
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#10(due in class on Thursday, April 17) Page 445, problems 2, 5, 6.
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#11(due in class on Thursday, April 24) Page 445, problems 1, 3, 4, 8, 9, 10.
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#12(due in class on Thursday, May 1) Let B_n denote the Artin braid group on n strands with generators b_1, b_2, . . . b_n-1 and relations b_j·b_i = b_i·b_j when |i−j| > 1, and b_j·b_i·b_j = b_i·b_j·b_i when j=i+1. See: http://en.wikipedia.org/wiki/Braid_group
What are the elements of finite order in B_n? (proof not required, just use intuition).
Show that the braid c_3 = b_1·b_2·b_1 satisfies c_3·b_j = (b_{3−j})·c_3 for j = 1,2. Conclude that (c_3)^2 lies in the center of B_3.
Show that the braid c_4 = b_1·b_2·b_3·c_3 satisfies c_4·b_j = (b_{4−j})·c_4 for j = 1,2,3. Conclude that (c_4)^2 lies in the center of B_4.
Find two subgroups H, K of B_n that commute with each other. Explain how to construct a Diffie-Hellman key exchange where g in B_n is public, Alice chooses an element a in H and Bob chooses b in K and they exchange the conjugates a·g·a^{-1}, b·g·b^{-1}. What is the shared secret? (This called the Ko-Lee Key Exchange).