Arithmetic and Algebra

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• ?????(Arithmetic and Algebra)
• ??(Combinatorics)
• ??(Number Theory)

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Last proof we didnt finish

• Optimal case of Quick Sort
• an2an/2 n , an?

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Number Theory and Divisibility

• The Study of the properties of the Integers,
  specifically integer divisibility.
• We say b divides a (denotes ba) if abk for some
  integer k.
• As a consequence of this definition, the smallest
  natural divisor of every non-zero integer is 1.
  In general there is no integer k such that a0k

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Prime Numbers

• A prime number is an integer p gt 1 which is only
  divisible by 1 and itself
• Every Integer can be expressed in only one way as
  the product of primes
• 1053x5x7
• 322x2x2x2x2
• Any number which is not prime is said to be
  composite

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Primality Testing and Factorization

• There are infinite number of primes (Euclids
  proof), but there are a lot of them.
• Roughly x/lnx primes less than or equal to x.
• The easiest way to find the prime factorization
  of an integer n is repeated division.
• The smallest prime factor is must at most square
  root of n unless n is prime.
• Each divisor is the product of some subset of
  these prime factors. Such subset can be
  constructed using backtracking techniques, but we
  must be careful about duplicate prime factors.
• Prime factorization if 12 is (2, 2 and 3) but 12
  has 6 divisors (1, 2, 3, 4, 6, 12)

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Greatest Common Divisor

• The largest divisor shared by a given pair of
  integers
• Euclid GCD algorithm rests on two observations
• If ba, then gcd(a,b)b
• If abtr for integer t and r, then
  gcd(a,b)gcd(b,r)
• Why? gcd(a,b)gcd(btr,b)
• Any common divisor of a and b must rest totally
  with r
• It can also find integers x and y such that
  axbygcd(a,b) which will prove quite useful in
  solving linear congruence

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Implementation

• / Find the gcd(p,q) and x,y such that px qy
  gcd(p,q) /
• long gcd(long p, long q, long x, long y)
• long x1,y1 / previous coefficients /
• long g / value of gcd(p,q) /
• if (q gt p) return(gcd(q,p,y,x))
• if (q 0)
• x 1
• y 0
• return(p)
• g gcd(q, pq, x1, y1)
• x y1

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Modular Arithmetic

• Sometimes computing the remainder is more
  important than a quotient
• What day will your birthday fall on next year?
• The key to such efficient computations is modular
  arithmetic
• RSA Diffie-Hellman
• What is (xy) mod n? We can simplify this to ((x
  mod n)(y mod n)) mod n to avoid adding big
  numbers.
• Subtraction is just a special case of addition
• (12 mod 100)-(53 mod 100) -41 mod 100 59 mod
  100

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Deffie-Hellman Algorithm

• Negotiate a common private key for end-to-end
  communication by exchanging a public key with
  each other
• Alice and Bob randomly generate their own private
  keys (XA and XB )
• They use their own private keys and two common
  numbers aand q to generate their public keys (YA
  and YB )
• Alice send public key YA to Bob, Bob send public
  key YB to Alice
• Alice use XA and YB to generate a key same with
  the key generated by Bobs XB and YA

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Proof by Number Theory
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Exponentiation

• Since multiplication is just repeated addition,
• xy mod n (x mod n)(y mod n) mod n
• Since exponentiation is just repeated
  multiplication
• xy mod n (x mod n)y mod n
• What is the last digit of 2100

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Congruences(??)

• Congruences are alternate notation for
  representing modular arithmetic.
• We say that ab(mod m) if m(a-b)
• The set of integers with a give remainder n
• What integers satisfy the congruence x3(mod 9)
• 9y3
• What about 2x3(mod 9) and 2x3(mod 4)

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Operations on Congruences

• Congruences support addition, subtract and
  multiplication, as well as a limited form of
  division-provided they share the same modulus
• Addition and Subtraction
• If ab(mod n) and cd(mod n) , then (ac)bd(mod
  n)
• 4x7(mod 9) and 3x3(mod 9), then 4x-3x7-3(mod
  9) ?x4(mod 9)
• Multiplication (general multiplication holds)
• ab(mod n) and cd(mod n) implies acbd(mod n).
• Division
• We cannot cavalierly cancel common factors from
  congruences, note that 62 61(mod 3), but
  clearly 2 ! (1 mod 3)

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Simplifying and Solving Congruences

• We can simplify a congruence adbd(mod dn) to
  ab(mod n)
• A linear congruence is an equation of the form
  axb(mod n), solving the x to satisfy the
  equation.
• ax1(mod n) has a solution if and only if
  gcd(a,n)1, i.e. axnygcd(a,n)1
• 3 cases to check possible solution
• gcd(a,b,n)gt1, then we can divide all three terms
  by this divisor to get an equivalent congruence
• gcd(a, n) does not divide b, no solution
• gcd(a, n)1, one solution, xa-1b

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More Advanced Tools

• Chinese remainder theorem
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• Diophantine equations
• xnynzn

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