Number Theory: Final Project. Camilla Hollanti, Mohamed Taoufiq Damir, Laia Amoros

Transcription

1 Number Theory: Final Project

2 0 About the Final Project (30 of the final grade) Projects can be realized individually or in pairs. Choose your topic and possible pair by Friday, November 16th: Link to Google Sheet The project consists of a short report (5-10 pages), to be submitted by Sunday, December 16th, and a short presentation ( 5 slides). Each group will give a 5-minute powerpoint presentation during Wednesday-Friday, December 5th-7th (Doodle later). With every project we suggest few references as a first reading, but it is not compulsory to follow a certain reference as long as you remain within scope. Number Theory: Final Project 2/16

3 Continued Fractions Give me a date (dd/mm/yyyy) and I ll give you which day of the week it was (Monday, Tuesday, etc). In fact, we have the following theorem: The n th day of the m th month (March counted as month number 1), in the year 100c + d, falls on the day [ ] 13m 1 W (n, m, c, d) = n + 5c + d + + [c/4] + [d/4] (mod 7), 5 where [x] stands for the integer part of x, and 1 = Monday, 2 = Tuesday, etc. One can prove the theorem above using the so called continued fractions. Tasks (References: [2], [9]) Give a short introduction to continued fractions. Explain how we can prove the theorem above using continued fractions. Number Theory: Final Project 3/16

4 The Prime Number Theorem The distribution of primes is still a big mystery as prime numbers seem to be randomly distributed. A natural question to ask is: for a given x, what is the number of primes less than x? An answer to this question was conjectured by Gauss and proved (approximately) by Hadamard and de La Vallée-Poussin. The Prime Number Theorem states that, asymptotically, the number of primes up to a given x is x logx. approximately Tasks (Reference: [1]) Explain roughly the steps of the proof (many alternatives exist). Explain the relation between the Prime Number Theorem and the Riemann hypothesis. Number Theory: Final Project 4/16

5 Primality Tests On December 26th, 2017, Jonathan Pace discovered the 50th known Mersenne prime 2 77,232, It is the largest prime number known to mankind. The question here is, how do we know that such a number really is a prime. Note that the number above has digits, so trying to test all the possible divisors will take an eternity. To this end, we need more sophisticated primality tests. Tasks (Reference: [11]) give an overview of the known primality tests. Explain how to prove that a Mersenne number is a prime, namely, the Lucas-Lehmer test. Number Theory: Final Project 5/16

6 Recurrent Sequences Recurrent sequences appear naturally in our daily lives, consequently having been used by mathematicians, physicists, and even artists. This project gives an overview of recurrent sequences with a focus on the Fibonacci sequence. Tasks (References: [10], [5]) give an overview of recurrent sequences with a focus on the appearance of Fibonacci sequence in our daily lives. Explain how to get the Binet Formula (the formula for the n th Fibonacci number). Calculate the complexity of the Euclidean algorithm we have seen on this course, namely, explain the worst case scenario of the Euclidean algorithm and its relation with the Fibonacci sequence. Number Theory: Final Project 6/16

7 Attacks on the RSA Crypto system RSA is the first example of a public key crypto system, and it has been implemented in several security systems. Unfortunately, it is also exposed to different attacks and cryptanalysis. Tasks (Reference: [3]) Give in detail at least five possible attacks on the RSA system. Number Theory: Final Project 7/16

8 Riemann Zeta Function Riemann Zeta function is one of the most studied objects in Analytic Number Theory. One of the important results related to this topic is the functional equation proved by Riemann, namely the analytic continuation of the zeta function to the complex plane. Tasks (References: [13], [15], [1, Chapter 12]) Highlight the important results related to the Riemann zeta function. Explain the steps leading to the functional equation. Some complex analysis background (e.g. Cauchy s integral theorem) is needed. Number Theory: Final Project 8/16

9 Sums of Squares Gauss and Lagrange proved that any integer n 0 can be expressed as a sum of squares n = x x 2 s for s {2, 4}. Tasks (Reference: [8, Chapter XX]) Explain how we can obtain the theorem above. Highlight the connection with the Gaussian integers Z[i] = {a + bi : a, b Z}, Hamiltonian quaternions, and lattices. Number Theory: Final Project 9/16

10 Private Information Retrieval In private information retrieval (PIR) a user wants to retrieve a file from a database, without revealing to the database (owner) which file she/he was after. The first computational PIR attempt was based on quadratic reciprocity, but turned out to be quite inefficient. Improvements exist. Tasks (References: [4], [7], [14]) Explain the concept of computational PIR. Give the details of the quadratic reciprocity based PIR scheme. Number Theory: Final Project 10/16

11 Introduction to Fermat s Last Theorem Consider the Diophantine equation x n + y n = z n with n 2. This equation has a non-trivial solution (x, y, z) Z 3 if and only if n = 2. This statement was conjectured by Fermat in 1639 and proved only after 358 years by Andrew Wiles in Tasks (References: [12], [8, Chapter 13]) Consider the case of Pythagorean triples (n = 2), Fermat s proof of the case n = 4, and Euler s proof of case n = 5. If you wish, you can attempt understanding the proof of Lamé and Kummer for regular primes. This involves linear combinations of the p-th roots of unity, similarly as in the Gauss sums (Stein, Ch. 4). Background in Abstract Algebra is helpful. This is also the main scope of our Algebraic Number Theory course! Number Theory: Final Project 11/16

12 Cryptography and Elliptic Curves During the course we will introduce RSA, a public key crypto system on Z n. Some cryptographic concepts have been extended to other groups, of which one of the most used one nowadays is the group of points on an elliptic curve over a finite field. Elliptic curves can be defined by an equation where a, b, c Z. Tasks (Reference: [6]) y 2 = x 3 + ax 2 + bx + c, Study this equation over the finite field Z p,and introduce elliptic curves based cryptography, as proposed by Koblitz and Miller in Show the so-called group law and introduce the discrete logarithm problem on elliptic curves. Number Theory: Final Project 12/16

13 0 References [1] T. Apostol. Introduction to Analytic Number Theory. Springer, [2] Sigrid Boege. Course notes. Continued Fractions, [3] Dan Boneh. Twenty years of attacks on the RSA cryptosystem. Notices of the AMS, 46( ), [4] M. Stadler C. Cachin, S. Micali. Computationally private information retrieval with polylogarithmic communication. Advances in Cryptology - EUROCRYPT 99, ( ), Number Theory: Final Project 13/16

14 0 References II [5] Ernie Croot. Notes on recurent sequences, [6] J. Bos et al. Elliptic curve cryptography in practice. IACR cryptology eprint archive, 2013(734), [7] Rafail Ostrovskyy. Eyal Kushilevitz. Replication is not needed: Single database, computationally-private information retrieval. In Proc. of the 38th Annu. IEEE Symp. on Foundations of Computer Science, (64-373), [8] E. M. Wright G. H. Hardy. An Introduction to the Theory of Numbers. Oxford University Press, Number Theory: Final Project 14/16

15 0 References III [9] Yury Grabovsky. Modern calendar and continued fractions. course notes, Temple University, [10] R. Honsberger. Mathematicals gems ii (chap 7). Dolciani Mathematical exposisions, [11] STEFAN LANCE. A survey of primality tests. Notes, [12] Reinhard Laubenbacher. voici ce que j ai trouvé: Sophie Germain s grand plan to prove fermat s last theorem. arxiv.org/pdf/ , Number Theory: Final Project 15/16

16 0 References IV [13] FELIX RUBIN. Riemann s first proof of the analytic continuation of ζ(s) and l(s, χ). In Seminar on Modular Forms, winter term, ETH zurich, [14] Valentina Settimi. A study of computational private information retrieval schemes and oblivious transfer. ALGANT Master thesis, [15] Andreas Steiger. Riemann s second proof of the analytic continuation of the riemann zeta function. In Seminar on Modular Forms, winter term, ETH zurich, Number Theory: Final Project 16/16