# THE UNIVERSITY OF SYDNEY Semester 2, Information Sheet for MATH2068/2988 Number Theory and Cryptography

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4 Objectives: The objectives of this unit are to: introduce basic concepts of number theory, such as primes and prime factorization, modular arithmetic, divisors and multiplicative functions, powers and discrete logarithms; explore standard proof techniques in number theory, such as induction and proof by contradiction; describe standard algorithms for computations in number theory, such as the Euclidean Algorithm and Pollard Rho algorithms; introduce the computer algebra package MAGMA and some of its number-theoretic functions; introduce basic concepts of cryptography, such as ciphering and deciphering, public and secret keys; describe classical cryptosystems such as substitution, transposition and block transposition ciphers; explore statistical attacks on classical cryptosystems; describe number-theoretic cryptosystems such as RSA, Elgamal, Diffie Hellman, Rabin s cryptosystem; introduce basic concepts of computational complexity, and use them to understand the relative effectiveness of cryptosystems; illustrate more difficult results in number theory. Outcomes: Students who successfully complete this unit should be able to: understand and use the basic terminology of number theory and cryptography; carry out simple number-theoretic computations either with a calculator or using MAGMA; apply standard number-theoretic algorithms; understand and use some classical and number-theoretic cryptosystems; apply standard methods to attack some classical cryptosystems; understand (see below) the theory underlying number-theoretic algorithms and cryptosystems, including the general properties of primes, prime factorization, modular arithmetic, divisors and multiplicative functions, powers and discrete logarithms. MATH2068 vs MATH2988: The main distinction in the expected outcomes between the mainstream unit MATH2068 and the advanced unit MATH2988 is in the depth of understanding of the underlying theory, both streams being expected to be familiar with the computational methods and algorithms described in the unit. MATH2068 students should be able to reproduce the proofs of the easier theorems in lectures and produce their own proofs of results at a similar level. MATH2988 students should be able to reproduce the proofs of the more difficult theorems in lectures (unless explicitly stated), produce their own proofs of results at a similar level, and devise their own algorithms to solve simple computational problems. Exercises in tutorials and computer labs are starred to indicate the level of difficulty. 4

5 Tentative week-by-week outline: Week Topics 1 Introduction, divisibility, prime and composite numbers Greatest common divisors, division algorithm, (extended) Euclidean algorithm Factorization: trial division and Fermat s method 2 Congruence notation, complete systems, reduced systems Fundamental Theorem of Arithmetic Inverses, powers and order in modular arithmetic 3 Basic concepts of cryptography, classical cryptosystems Statistical attacks on classical cryptosystems Euler Fermat Theorem, Fermat s Little Theorem 4 Relating congruences with different moduli Chinese Remainder Theorem, more on powers (Not for assessment) The Data Encryption Standard cryptosystem 5 Multiplicative functions, Euler s phi function Sum and number of divisors, perfect numbers 6 Relating different multiplicative functions Möbius inversion formula The RSA public key cryptosystem 7 Computational complexity, bit operations Big-O notation, polynomial-time algorithms Computational complexity of Euclidean and power algorithms 8 Pollard s Rho factorization algorithm Polynomial congruences Powers and primitive roots modulo a prime 9 Diffie Hellman key exchange protocol, Elgamal cryptosystem Safe primes, discrete logarithms Applications of primitive roots 10 Modular Lagrange interpolation formula and secret sharing Baby-step/giant-step and Pohlig Hellman algorithm for discrete logarithms 11 More on computing discrete logarithms Square roots modulo a prime Rabin s public key cryptosystem 12 (Not for assessment) Quadratic reciprocity theorem and other topics 13 Revision 5

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