Mathematical Background

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Mathematical Background
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Content

• Probability theory
• Complexity Theory
• Number theory
• Abstract Algebra

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Further readings

• General Birthday problem
• Information theory (entropy, )

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Probability theory

• Bayes theorem

• Event E, sample space S.
• 0 lt P(E) lt 1.
• P(E) E/S

• Birthday paradox
• The probability that at least 2 people in a room
  of 23 people have the same birthday is P(365,23),
  approximate 0.507

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Algorithm Complexity

• Basic definitions
• Algorithm, size of input
• Running time worst-case, average-case
• Asymptotic notations

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• Properties of order notation
• Increasing order of asymptotic growth rates

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Complexity classes

• Decision problems are problems which have either
  Yes or No as an answer.

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Further readings

• Some NP problems
• NP-hard, NP-complete
• Randomized algorithms

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Number theory
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Primes
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Factorization
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Euler phi function
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Algorithms in Z
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Extended Euclidean algorithm
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The integers modulo n
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The Zn
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Chinese remainder theorem
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The group Zn
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Orders of elements in Zn
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Quadratic residue
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Algorithms in Zn

• Basic arithmetics
• Addition, subtraction, multiplication

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Further readings

• The Legendre and Jacobi symbols

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Abstract Algebra

• Groups
• Group operations, finite, order, commutative
• Subgroup, cyclic
• Lagranges theorem
• Rings
• Fields
• Characteristic
• Subfield, field extension

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Polynomial rings Znx

• Irreducible polynomial
• Fx /(f(x))
• Gcd ,Computing multiplicative inverse
• Fast exponentiation

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Finite Fields
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Further readings

• Euclidean algorithms and arithmetic for
  polynomials in finite fields