Michel Waldschmidt

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October 15, 2005
Khemarak University, Phnom Penh
Explosion of Mathematics

• Michel Waldschmidt
• Université P. et M. Curie Paris VI
• Société Mathématique de France

http//www.math.jussieu.fr/miw
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• Lexplosion
• des
• Mathématiques

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http//smf.emath.fr/Publication/ExplosionDesMathe
matiques/ Presentation.html
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Explosion of Mathematics

• Weather forecast
• Cell phones
• Cryptography
• Control theory
• From DNA to knot theory

• Air transportation
• Internet modelisation of traffic
• Communication without errors
• Reconstruction of surfaces for images

Société Mathématique de France Société de
Mathématiques Appliquées et Industrielles
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Aim To illustrate with a few examples the
usefulness of some mathematical theories which
were developed only for theoretical
purposes Unexpected interactions between pure
research and the real world .
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Interactions between physics and mathematics

• Classical mechanics
• Non-Euclidean geometry
  Bolyai, Lobachevsky, Poincaré, Einstein
• String theory
• Global theory of particles and their
  interactions geometry in 11 dimensions?

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Eugene Wigner

•  The unreasonable effectiveness
• of mathematics in the natural
• sciences 
• Communications in Pure and Applied
  Mathematics, vol. 13, No. I (February 1960)

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Dynamical systems Three body problems
(Henri Poincaré) Chaos theory (Edward Lorentz)
the butterfly effect Due to
nonlinearities in weather processes, a butterfly
flapping its wings in Tahiti can, in theory,
produce a tornado in Kansas.
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Weather forecast Probabilistic model for
the climate Stochastic partial differential
equations Statistics
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Weather forecast

• Mathematical models are required for describing
  and understanding the processes of meteorology,
  in order to analyze and understand the mechanisms
  of the climate.
• Some processes in meteorology are chaotic, but
  there is a hope to perform reliable climatic
  forecast.

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Knot theory in algebraic topology

• Classification of knots, search of invariants
• Surgical operations

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Knot theory and molecular biology

• The topology of DNA molecule has an action on its
  biological action.
• The surgical operations introduced in algebraic
  topology have biochemical equivalents which are
  realized by topoisomerases.

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Finite fields and coding theory

• Solving algebraic equations with
  radicals Finite fields theory
  Evariste Galois
  (1811-1832)
• Construction of regular polygons with rule and
  compass
• Group theory

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Error Correcting Codes Data Transmission

• Telephone
• CD or DVD
• Image transmission
• Sending information through the Internet
• Radio control of satellites

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• Olympus Mons on Mars Planet
• Image from Mariner 2 in 1971.

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Sphere packing
The kissing number is 12
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Sphere Packing

• Kepler Problem maximal density of a packing of
  identical sphères
•   p / Ö 18 0.740 480 49
• Conjectured in 1611.
• Proved in 1999 by Thomas Hales.
• Connections with crystallography.

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Codes and Geometry

• 1949 Golay (specialist of radars) efficient code
  
• Eruptions on Io (Jupiters volcanic moon)
• 1963 John Leech uses Golays ideas for sphere
  packing in dimension 24 - classification of
  finite simple groups

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Data transmission

• French-German war of 1870, siege of Paris

Flying pigeons first crusade - siege of Tyr,
Sultan of Damascus
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Data transmission

• James C. Maxwell
• (1831-1879)
• Electromagnetism

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Cell Phones Information
Theory Transmission by Hertz waves Algorithmic,
combinatoric optimization, numerical treatment
of signals, error correcting codes. How to
distribute frequencies among users.
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Data Transmission
Transmission
Source Receiver
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Language Theory

• Alphabet - for instance 0,1
• Letters (or bits) 0 and 1
• Words (octets - example 0 1 0 1 0 1 0 0)

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ASCII

• American Standard Code for Information
  Interchange
• Letters octet
• A 01000001
• B 01000010

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Coding
transmission
Source Coded Text Coded Text Receiver
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Cryptography
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Encryption for security
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Applications of cryptography

• Credit cards
• Web security
• Imaging
• Encrypted television,
• Telecommunications

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Mathematics in cryptography

• Algebra
• Arithmetic, number theory
• Geometry

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History

• Encryption using alphabetical transpositions and
  substitutions (Julius Caesar).
• 1586, Blaise de Vigenère (key table
  of Vigenère)
• 1850, Charles Babbage (frequency of
  occurrences of lettres)

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Interpretation of hieroglyphs

• Jean-François Champollion (1790-1832)
• Rosette stone (1799)

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Any secure encyphering method is supposed to be
known by the ennemy The security of the system
depends only on the choice of
keys.

• Auguste Kerckhoffs
• La  cryptographie militaire,
• Journal des sciences militaires, vol. IX,
• pp. 538, Janvier 1883,
• pp. 161191, Février 1883 .

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• 1917, Gilbert Vernam (disposable mask)
• Example the red phone Kremlin/White House
• 1940, Claude Shannon proves that the only secure
  private key systems are those with a key at least
  as long as the message to be sent.

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Enigma
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Alan Turing

• Deciphering coded messages (Enigma)
• Computer science

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Colossus

• Max Newman,
• the first programmable electronic computer
  (Bletchley Park before 1945)

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Théorie de lInformation

• Claude Shannon
• A mathematical theory of communication
• Bell System Technical Journal, 1948.

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• Claude E. Shannon, " Communication Theory of
  Secrecy Systems ", Bell System Technical Journal
  , vol.28-4, page 656--715, 1949. .

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DES Data Encryption Standard

• In 1970, NBS (National Board of Standards)
  put out a call in the Federal Register for an
  encryption algorithm
• with a high level of security which does not
  depend on the confidentiality of the algorithm
  but only on secret keys
• using secret keys which are not too large
• fast, strong, cheap
• easy to implement
• DES was approved in 1978 by NBS

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Algorithm DEScombinations, substitutions and
permutations between the text and the key

• The text is split in blocks of 64 bits
• The blocks are permuted
• They are cut in two parts, right and left
• Repetition 16 times of permutations and
  substitutions
• One joins the left and right parts and performs
  the inverse permutations.

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Diffie-Hellmancryptography with public key

• W. Diffie and M.E. Hellman,
• New directions in cryptography,
• IEEE Transactions on Information
  Theory,
• 22 (1976), 644-654

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RSA (Rivest, Shamir, Adleman - 1978)
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R.L. Rivest, A. Shamir, and L.M. Adleman,

• A method for obtaining digital signatures and
  public-key cryptosystems,
• Communications of the ACM
• (2) 21 (1978), 120-126

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Public key

• Easy to multiply two numbers, even if they are
  large.
• If you know only the product, it is difficult to
  find the two numbers.

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Example

• p111395432514882798792549017547702484407092284484
  3
• q191748170252450443937578626823086218069693418929
  3
• pq21359870359209100823950227049996287970510953418
  26417406442524165008583957746445088405009430865999
  

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• Quizz du malfaiteur
• Apprenez les maths
• pour devenir chef du Gang
• http//www.parodie.com/monetique/hacking.htm
• http//news.voila.fr/news/fr.misc.cryptologie

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Primality tests

• Given an integer, decide whether it is the
  product of two smaller numbers or not.
• 8051 is composite
• 805183 ?97, 83 are 97 prime
• Todays limit more than 1000 digits

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Industrial primes

• Probabilistic Tests are not genuine primality
  tests they do not garantee that the given number
  is prime. But they are useful whenever a small
  rate or error is allowed. They produce the
  industrial primes.

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Agrawal-Kayal-Saxena

• Manindra Agrawal, Neeraj Kayal and Nitin Saxena,
  PRIMES is in P
• (July 2002)

http//www.cse.iitk.ac.in/news/primality.html
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Largest known primes
213 466 917 -1
November 14, 2001
4 053 946 digits
220 996 011 -1
November 17, 2003
6 320 430 digits
http//primes.utm.edu/largest.html
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• The 3 largest known primes can be written as 2a
  -1 (and we know 40 such primes)
• We know
• 4 primes with more than 1 000 000 digits,
• 13 primes with more than 500 000 digits,
• 94 primes with more than 200 000 digits

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Mersenne numbers (1588-1648)

• Mersenne numbers are numbers of the form Mp2p
  -1 with p prime.
• 22 944 999 -1 is composite divisible by
  314584703073057080643101377

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Perfect numbers

• An integer n is called perfect if n is the sum of
  the divisors of n distinct from n.
• The divisors of 28 distinct from 28 are
  1,2,4,7,14 and 28124714.
• Note that 284 ? 7 while 7M3.

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Even perfect numbers (Euclid)

• Even perfect numbers are numbers which can be
  written 2p-1 ? Mp with Mp 2p -1 a Mersenne
  prime (hence p is prime).
• Are-there infinitely many perfect numbers?
• Nobody knows whether there exists any odd perfect
  numbers.

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Fermat numbers (1601-1665)

• A Fermat number is a number which can be written
  Fn22n1.
• Construction with rule and compass of regular
  polygons.
• F15, F2 17, F3257, F465537 are prime numbers

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Euler(1707-1783)

• F5 2321 is divisible by 641
• 4 294 967 297 641 ? 6 700 417
• 641 54 24 5 ? 27 1
• Are there infinitely many Fermat Primes?

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John Cosgrave February
2003 Fermat number 222 145 352 1 is divisible
by 3?22 145 353 1, a prime number with 645
817 digits October 12, 2003 Fermat number 222
478 782 1 is divisible by 3 ? 22 478 785 1 a
prime number with 746 190 digits
www.spd.dcu.ie/johnbcos
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Factorization algorithms

• Given a prime number, decompose it into a product
  of prime numbers
• Todays limit around 150 digits
• http//www.rsasecurity.com/rsalabs/challenges/

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Challenge Number Prize US

• RSA-576 10,000 Not Factored   
• RSA-640 20,000 Not Factored   
• RSA-704 30,000 Not Factored   
• RSA-768 50,000 Not Factored   
• RSA-896 75,000 Not Factored   
• RSA-1024 100,000 Not Factored   
• RSA-1536 150,000 Not Factored   
• RSA-2048 200,000 Not Factored   

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RSA-576 Prize 10,000 Status Not Factored
Decimal Digits 174

• 18819881292060796383869723946165043980716356337941
  73827007633564229888597152346654853190606065047430
  45317388011303396716199692321205734031879550656996
  221305168759307650257059
• Digit Sum 785   

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21024 1 45592577 ? 6487031809
? 4659775785220018543264560743076778192897 ?
p252 http//discus.anu.edu.au/r
pb/F10.html
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Integers modulo n

• Examples
• n2 (odd and even numbers)
• n7 (days of the week)
• n9 (sum of digits)
• n10 (keep only the last digit)
• n12 (Zodiac)
• n24 (hours)
• n60 (minutes, seconds)

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Powers modulo 1000

• Example modulo 1000, keep only the last 3
  digits
• Start with x111 (odd and not multiple of 5 last
  digit is 1, 3, 7 or 9)
• x2 12 321 write x2 ? 321 mod 1000
• 111 ? 321 35 631 hence x3 ? 631 mod 1000
• Assume you know x3 ? 631 mod 1000 - how to
  recover x ?
• There are 400 integers less than 1 000 which are
  odd and not multiple of 5.

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Powers modulo 1000 (continued)

• Continue to compute x4 ? 41 mod 1000
• x8 ? 681 mod 1000
• x16 ? 761 mod 1000
• x32 ? 121 mod 1000
• x64 ? 641 mod 1000
• x67 ? x64 ? x3 ? 641 ? 631 ? 471 mod 1000
• x201 ? (x67)3 ? 111 ? x mod 1000

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Powers modulo 1000 (again)

• The theory predicts that
• x200 ? 1 mod 1000
• hence
• x201 ? x mod 1000, x401 ? x mod 1000,
• In fact computations show that already
• x100 ? 1 mod 1000

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Retreive x from x 3 modulo 1000

• We know x3 ? 631 mod 1000
• Also x201 ? x mod 1000.
  Notice that 20167 ? 3
• Hence x ? (x3)67 mod 1000
• Compute 63167 ? 111 mod 1000
• Therefore x ? 111 mod 1000

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Retreive x from x 7 modulo 1000

• It works if 3 is replaced by any odd number not
  divisible by 5 (last digit is 1, 3, 7 or 9)
• For instance with 7 use 1001143 ? 7.
• If you know x7 ? 871 mod 1000 then compute
  
• 871143 ? 111 mod 1000
• therefore
• x ? x1001 ? 111 mod 1000

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Message modulo n

• Fix a positive integer n this is the size of
  the messages which are going to be sent.
• All computation will be done modulo n we
  replace each integer by the remainder in its
  division by n.

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Size of n

• We take for n the product of two prime numbers
  with some 150 digits each.
• The product has some 300 digits computers cannot
  find the two prime numbers.

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Public key cryptography

• Public key (e,n)
• e and n are integers
• n gives the size of messages
• e is used for encryption.
• Private key f an integer, used for decryption,
  known only by receiver.

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Choice of e, f and n

• Select two large prime numbers p and q
• Take npq.
• Next select e and f such that
• xef ? x mod n
• for all x not divisible by p nor by q.

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Cryptography with public key

• The message to be sent is an integer x with 0 lt x
  lt n
• The sender sends y ? xe mod n
• The receiver computes z ? yf mod n
• Because of the choice of e and f, a miracle
  occurs
• z ? x mod n.

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Explanation of z ? x mod n

• Fermats little theorem If p is prime, then
  for any positive integer x,
• xp ? x mod p
• If you know e, p and q it is easy to find f such
  that
• xef ? x mod n
• (condition e and (p-1)(q-1) have no common
  factor)

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Finding f

• Condition on e and f
• ef ? 1 mod (p-1)(q-1)
• To compute f when one knows e, one needs to
  factorize p-1 and q-1.
• If one knows e and n but not p and q, one is not
  able so far to find f.

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Auxiliary questions

• Send the key
• Identify the sender certification of signatures

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Sending a suitcase

• Assume Alice has a suitcase and a locker she
  wants to send the suitcase to Bob in a secure way
  so that nobody can see the content of the
  suitcase.
• Bob also has a locker and the corresponding key,
  but they are not compatible with Alices ones.

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The protocol of the suitcases

• Alice closes the suitcase with her locker and
  send it to Bob.
• Bob puts his own locker and sends back the
  suitcase with two lockers.
• Alice removes her locker and sends back the
  suitcase to Bob.
• Finally Bob is able to open the suitcase.

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Mathematical translation

• The suitcase is replaced by the message x, the
  lockers by public keys e, the keys of the lockers
  by private keys f.
• Alice has a public key eA and a private key fA .
• Bob also has a public key eB and a private key
  fB .
• Alice sends to Bob x eA modulo n.
• Bob sends back (x eA ) eB x eA eB modulo n.
• Alice sends to Bob (x eA eB ) fA x eB modulo
  n.
• Finally Bob computes (x eB ) fB x modulo n.

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Security of bank cards
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Secret code of a bank card
The memory electronic card (chip card) was
invented in the 70s by two french engineers,
Roland Moreno and Michel Ugon.

• You need to identify yourself to the bank. You
  know your secret code f, but for security reason
  you are not going to send it to the bank.
  Everybody (including the bank) knows the public
  key e. Only you know f.

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Secret code of a bank card

• The bank sends a random message x
• You send back yx f modulo n.
• The bank computes ye x fe modulo n and checks
  that it is x modulo n.

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Modern cryptography

• Elliptic curves (discrete logarithm)
• Jacobian of algebraic curves
• Quantum cryptography (Peter Shor) - magnetic
  nuclear resonance

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Error correcting codes
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Applications of error correcting codes

• Transmitions by satellites
• Compact discs
• Cellular phones

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Codes and Maths

• Algebra
• (discrete mathematics finite fields, linear
  algebra,)
• Geometry
• Probability and statistics

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Coding
transmission
Source Coded Text Coded Text Receiver
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Coding
transmission
Source Coded Text Noise CodedText Receiver
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• Principle of coding theory
• only certain words are permitted (code
  dictionary of allowed words).
• The  useful  letters carry the information,
  the other ones (control bits) allow detecting
  errors.

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Detecting one error

• Send twice the same message
• 2 code words on 422
• (1 useful letter of 2)

• Code words
• (two letters)
• 0 0
• 1 1
• Rate 1/2

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Correcting an error

• Send the same message three times
• 2 code words of 823
• (1 useful letter of 3)

• Code words
• (three letters)
• 0 0 0
• 1 1 1
• Rate 1/3

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• Correct 0 0 1 as 0 0 0
• 0 1 0 as 0 0 0
• 1 0 0 as 0 0 0
• and
• 1 1 0 as 1 1 1
• 1 0 1 as 1 1 1
• 0 1 1 as 1 1 1

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• Principle of coding correcting one error
• Two distinct code words have at least three
  distinct letters

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Detecting one error (again)

• Code words (three letters)
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0
• Parity bit (x y z) with zxy.
• 42?22 code words of 823
• (2 useful letters of 3).
• Rate 2/3

2
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Code words Non code words

• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0

• 0 0 1
• 0 1 0
• 1 0 0
• 1 1 1

• Two distinct code words have at least
  two distinct letters.

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Correcting one error (again)

• Words of 7 letters
• Code words (1624 on 12827 )
• (a b c d e f g)
• with
• eabd
• facd
• gabc
• Rate 4/7

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How to compute e , f , g , from a , b , c , d.
eabd
d
a
b
facd
c
gabc
100
16 code words of 7 letters

• 0 0 0 0 0 0 0
• 0 0 0 1 1 1 0
• 0 0 1 0 0 1 1
• 0 0 1 1 1 0 1
• 0 1 0 0 1 0 1
• 0 1 0 1 0 1 1
• 0 1 1 0 1 1 0
• 0 1 1 1 0 0 0

• 1 0 0 0 1 1 1
• 1 0 0 1 0 0 1
• 1 0 1 0 1 0 0
• 1 0 1 1 0 1 0
• 1 1 0 0 0 1 0
• 1 1 0 1 1 0 0
• 1 1 1 0 0 0 1
• 1 1 1 1 1 1 1

Two distinct code words have at least three
distinct letters
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Listening to a CD

• On a CD as well as on a computer, each sound is
  coded by a sequence of 0s and 1s, grouped in
  octets
• Further octets are added which detect and correct
  small mistakes.

102
Coding the sound on a CD

• Using a finite field with 256 elements, it is
  possible to correct 2 errors in each word of 32
  octets with 4 control octets for 28 information
  octets.

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A CD of high quality may have more than 500
000 errors!

• 1 second of radio signal 1 411 200 bits.
• The mathematical theory of error correcting codes
  provides more reliability and at the same time
  decreases the cost. It is used also for data
  transmission via the internet or satellites

104

• Informations was sent to the earth using an
  error correcting code which corrected 7 bits on
  32.
• In each group of 32 bits, 26 are control bits
  and the 6 others contain the information.

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Voyager 1 and 2 (1977)

• Journey Cape Canaveral, Jupiter, Saturn, Uranus,
  Neptune.
• Sent information by means of a binary code which
  corrected 3 errors on words of length 24.

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Mariner spacecraft 9 (1979)

• Sent black and white photographs of Mars
• Grid of 600 by 600, each pixel being assigned one
  of 64 brightness levels
• Reed-Muller code with 64 words of 32 letters,
  minimal distance 16, correcting 7 errors, rate
  3/16

107
Voyager (1979-81)

• Color photos of Jupiter and Saturn
• Golay code with 4096212 words of 24 letters,
  minimal distance 8, corrects 3 errors, rate 1/2.
• 1998 lost of control of Soho satellite recovered
  thanks to double correction by turbo code.

108
The binary code of Hamming and Shannon (1948)

• It is a linear code (the sum of two code words
  is a code word) and the 16 balls of radius 1 with
  centers in the code words cover all the space of
  the 128 binary words of length 7
• (each word has 7 neighbors (71)?16 256).

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The Hat Problem

• A team of three people in a room with black/white
  hats on their head (hat colors chosen at random).
  Each of them sees the color on the hat of the
  others but not on his own. They do not
  communicate.
• Everyone writes on a piece of paper the color he
  guesses for his own hat black/white/abstain

110

• The team wins if at least one of the three people
  does not abstain, and everyone who did not
  abstain guesses correctly the color of his hat.

111

• Simple strategy they agree that two of them
  abstain and the other guesses. Probability of
  winning 1/2.
• Is it possible to do better?

112

• Hint
• Improve the probability by using the available
  information each member of the team knows the
  two other colors.

113

• Better strategy if a member sees two different
  colors, he abstains. If he sees the same color
  twice, he guesses that his hat has the other
  color.

114

• Wins!

115

• Loses!

116

• Winning

117

• Losing

118

• The team wins exactly when the three hats do not
  have all the same color, that is in 6 cases of a
  total of 8
• Probability of winning 3/4.

119

• Are there better strategies?
• Answer NO!
• Are there other strategies giving the same
  probability 3/4?
• Answer YES!

120
Tails and Ends

• Throw a coin three consecutive times
• There are 8 possible sequences of results
• (0,0,0), (0,0,1), (0,1,0), (0,1,1),
• (1,0,0), (1,0,1), (1,1,0), (1,1,1).

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If you bet (0,1,0), you have

• All three correct results for (0,1,0).
• Exactly two correct results if the sequence is
  either (0,1,1), (0,0,0) or (1,1,0),
• Exactly one correct result if the sequence is
  either (0,0,1), (1,1,1) or (1,0,0),
• No correct result at all for (1,0,1).

122
Whatever the sequence is, among 8 possibilities,

• each bet
• is winning in exactly 1 case
• has exactly two correct results in 3 cases
• has exactly one correct result in 3 cases
• has no correct result at all in only 1 case

123

• Goal To be sure of having at least two correct
  results
• Clearly, one bet is not sufficient
• Are two bets sufficient?
• Recall that there are 8 possible results, and
  that each bet has at least two correct results in
  4 cases.

124
Answer YES, two bets
suffice!

• For instance bet
• (0,0,0) and (1,1,1)
• Whatever the result is, one of the two digits
• 0 and 1
• occurs more than once.
• Hence one (and only one) of the two bets
• has at least two correct results.

125
Other solutions

• Select any two bets with all three different
  digits, say
• 0 0 1 and 1 1 0
• The result either is one of these, or else has
  just one common digit with one of these and two
  common digits with the other.

126

• Come back with
• (0,0,0) and (1,1,1)
• The 8 sequences of three digits
• 0 and 1
• split into two groups
• those with two or three 0s
• and
• those with two or three 1s

127
Hamming Distance between two words

• number of places where the two words
• do not have the same letter
• Examples
• (0,0,1) and (0,0,0) have distance 1
• (1,0,1) and (1,1,0) have distance 2
• (0,0,1) and (1,1,0) have distance 3
• Richard W. Hamming (1915-1998)

128
Hamming Distance

• Recall that the Hamming distance between two
  words is the number of places where letters
  differ.
• A code detects n errors iff the Hamming distance
  between two distinct code words is at least 2n.
• It corrects n errors iff the Hamming distance
  between two distinct code words is at least
  2n1.

129

• The set of eight elements splits into two balls
• The centers are (0,0,0) and (1,1,1)
• Each of the two balls consists of elements at
  distance at most 1 from the center

130
Back to the Hat Problem

• Replace white by 0 and black by 1
• hence the distribution of colors becomes a
  word of three letters on the alphabet 0 , 1
• Consider the centers of the balls (0,0,0) and
  (1,1,1).
• The team bets that the distribution of colors is
  not one of the two centers.

131
Assume the distribution of hats does not
correspond to one of the centers (0, 0, 0) and
(1, 1, 1). Then

• One color occurs exactly twice (the word has both
  digits 0 and 1).
• Exactly one member of the team sees twice the
  same color this corresponds to 0 0 in case he
  sees two white hats, 1 1 in case he sees two
  black hats.
• Hence he knows the center of the ball (0, 0, 0)
  in the first case, (1, 1, 1) in the second case.
• He bets the missing digit does not yield the
  center.

132

• The two others see two different colors, hence
  they do not know the center of the ball. They
  abstain.
• Therefore the team win when the distribution of
  colors does not correspond to the centers of the
  balls.
• this is why the team win in 6 cases.

133

• Now if the word corresponding to the distribution
  of the hats is one of the centers, all members of
  the team bet the wrong answer!
• They lose in 2 cases.

134
Another strategy

• Select two words with mutual distance 3
  two words with three distinct letters, say
  (0,0,1) and (1,1,0)
• For each of them, consider the ball of elements
  at distance at most 1

135

• (0,0,0)
• (0,0,1) (0,1,1)
• (1,0,1)

• (1,1,1)
• (1,1,0) (1,0,0)
• (0,1,0)

136

• The team bets that the distribution of colors is
  not one of the two centers (0,0,1), (1,1,0) .
• A word not in the center has exactly one letter
  distinct from the center of its ball, and two
  letters different from the other center.

137
Assume the word corresponding to the distribution
of the hats is not a center. Then

• Exactly one member of the team knows the center
  of the ball. He bets the distribution does not
  correspond to the center.
• The others do not know the center of the ball.
  They abstain.
• Hence the team win.

138
The Hat Problem with 7 people

• The team bets that the distribution of the hats
  does not correspond to the 16 elements of the
  Hamming code
• Loses in 16 cases (they all fail)
• Wins in 128-16112 cases (one bets correctly, the
  6 others abstain)
• Probability of winning 112/1287/8

139
Tossing a coin 7 times

• There are 128 possible results
• Each bet is a word of 7 letters on the alphabet
  0, 1
• How many bets do you need if you want to
  guarantee at least 6 correct results?

140

• Each of the 16 code words has 7 neighbors (at
  distance 1), hence the ball of which it is the
  center has 8 elements.
• Each of the 128 words is in exactly one of these
  balls.

141

• Make 16 bets corresponding to the 16 code words
  then, whatever the result is, exactly one of
  your bets will have at least 6 correct answers.

142
The price of financial options

• Probability theory yields a modelisation of
  random processes. The prices of stocks traded on
  stock exchanges fluctuate like the Brownian
  motion.
• Optimal stochastic control involves ideas which
  previously occurred in physics and geometry
  (deformation of surfaces).

143
How to control a complex world

• Managing distribution in an electricity network,
  studying the vibrations of a bridge, the flow of
  air around an airplane require tools from the
  mathematical theory of control (differential
  equations, partial derivatives equations) .
• The optimization of trajectories of satellites
  rely on optimal control, numerical analysis,
  scientific calculus,

144
Optimization

• Industry manufacturing, costs reducing,
  decreasing production time,
• Production of fabrics, shoes minimizing waste,
  
• Petroleum Industry how to find the proper
  hydrocarbon mixtures,
• Aero dynamism (planes, cars,).
• Aerospace industry optimal trajectory of an
  interplanetary spaceflight,

145
Mathematics involved in optimization

• Algebra (linear and bilinear algebra,)
• Analysis (differential calculus, numerical
  analysis, )
• Probability theory.

146
Optimal path

• How to go from O to F

B
A
c
a
d
C
b
y
x
F
O
D
z
t
E
af(x1,,xn)
147
Trees and graphs

• A company wants to find the best way (less
  expensive, fastest) for trucks which receive
  goods and deliver them at many different places.

148
Applications of graph theory

• Electric circuits
• How to rationalize the production methods, to
  improve the organization of a company.
• How to manage the car traffic or the metro
  network.
• Informatics and algorithmic
• Buildings and public works
• Internet, cell phones

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