Mathematics Building, Room 422
Columbia University
New York, NY 10027 
(212) 854-4304
       Fax: (212) 854-8962


Research Interests:  Number Theory curriculum vitae
Some Preprints: (pdf files)
An additive prime divisor function of Alladi and Erdős
A standard zero free region for Rankin-Selberg L-functions (with X. Li)

A converse theorem for double Dirichlet series and Shintani zeta functions
                                                   (with N. Diamantis)
On the determination of the Plancherel measure for Lebedev-Whittaker transforms on GL(n)
(with A. Kontorovich)
Fourier expansions of GL(2) newforms at various cusps
                                        (with J. Hundley and M. Lee)
Natural boundaries and the correct notion of integral moments of L-functions
                                                            (with A. Diaconu and P. Garrett)
Moments for L-functions for GL(r) x GL(r-1) (with A. Diaconu and P. Garrett)
A converse theorem for double Dirichlet series (with N. Diamantis)
Voronoi Formulas on GL(n) (with X. Li)
Second Moments of GL(2) L-Functions (with A. Diaconu)
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 

Automorphic Representations and L-Functions for the General Linear Group, Volume 1
Automorphic Representations and L-Functions for the General Linear Group, Volume 2
Automorphic Forms and L-Functions for the Group GL(n,R) 
Blog with corrections of Errors in the above books
                    (Goldfeld Conference photos)

                    Joint COLUMBIA-CUNY-NYU Number Theory Seminar
                    Bretton Woods Workshop on Multiple Dirichlet Series(2005)    (Conference photos)
                    AAG key exchange
                    Edinburgh Workshop on Multiple Dirichlet Series(2008)
                    Decision Regarding World Record Musky Challenge
                    My Daughter's art work (including my portrait above)
                    Some Interesting Chemistry
                    Very accurate clock


  TR 2:40-3:55  Location: Room 312 Mathematics Building

Introduction to Cryptography with Coding Theory (Second Edition)
                    by Wade Trappe and Lawrence Washington.

TA:  James Cornish

Help Room (Math 406) Hours: 
Wed. 12-1, Thurs. 1-3.

Jaewan Bahk:  Help Room Hours: Monday 12-2.
Bharatha Rankothge:  Help Room Hours: Tuesday 10-12.

MIDTERM EXAM 1:  Tuesday October 3 in class 2:40-3:55pm (20% of total grade for this class).

MIDTERM EXAM 2:  Tuesday, November 21 in class 2:40-3:55pm  (20% of total grade for this class).

FINAL EXAM:  Thursday,  December 21 in class 1:10-4pm (35% of total grade for this class).

HOMEWORK (25% of total grade for this course) 
Please put all homework in the box just opposite Room 415 Math Building.
Deadline is Thursday 11:00pm (Note that Math Building is locked by Facilities at 11:00pm).
All homework is from the Trappe-Washington book.

HW#1 due Thursday, September 14:  Page 55, Problems 2, 3, 4, 5, 8, 10, 11

HW#2 due Thursday, September 21:  Page 104, Problems 1, 2, 3, 5, 12, 13, 18, 19, 26

HW#3 due  Thursday, September 28:  Page 105, Problems 9, 10  Page 192, Problems 1, 3, 7, 9   Page 214, Problems 1, 2, 11

HW#4 due Thursday, October 12: ATTACKS ON RSA AND DIFFIE-HELLMAN
Problem 1:  Using the fact that 1024 = (2^6)x(2^3)x(2)
compute  3^1024 (mod 7). (See section 3.5, Modular Exponentiation). 
Problem 2: Factor 497 using the Pollard p-1 Factoring Algorithm with B! = 3! and a=3.
Notes on Pollard's p-1 Attack
POHLIG-HELLMAN ATTACK: Page 215, Problems 3, 4.  Notes on the Pohlig-Hellman Attack
Page 286, Problem 4 (see section 10.1.1 Intruder -in-the-Middle Attacks).

HASH FUNCTIONS: Page 239, Problems 1, 2 see--> SquareRoots(mod pq) ,
Page 239, Problem 4 (see section 8.4 Birthday Attacks)    Birthday Attack
EL GAMAL SIGNATURES: Page 252, Problems 1, 2, 4, 5, 8.

HW#6 due Thursday, October 26: ELLIPTIC CURVE CRYPTOGRAPHY
Page 370, Problems 1, 2, 3, 4, 5, 10  Notes on Elliptic Curve Key Exchange
HW#7 due Thursday, November 2:
COIN FLIPPING: Page 315, Problems 2, 3   SquareRoots(mod p)
ZERO KNOWLEDGE TECHNIQUES: Page 321, Problems 1, 2, 3, 4, 5, 6   Zero Knowledge Proofs in a Nutshell
                                How to Explain Zero Knowledge Protocols to Your Children

HW#8 due Thursday, November 9:  
IRREDUCIBLE POLYNOMIALS Page 110, Problems 33, 34,  Page 447,  Problem 12(a).
FINITE FIELDS: Construct a finite field of 8 elements.

HW#9 due Thursday, November 16: ERROR CORRECTING CODES
Page 445, Problems 1,  2,  3(a),(b),(c)

MIDTERM EXAM #2 Tuesday, Nov. 21

HW#10 due Thursday, November 30:  BINARY CODES
Page 445, Problems  4,  5(a),(b),  6, 8

HW#11 due Thursday, December 7: LINEAR CODES
Page 445, Problems  9, 10, 11, 12, 13