Number theory
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------- Prime numbers

Prime number
  Definition: An integer whose only factors are 1 and itself
  pi(n) is the number of primes <n.
    lim(n->inf) pi(n) / (n/ln n) = 1, so any large number n has about
    1 in ln(n) chance of being prime.

Prime factorization, AKA the Fundamental Theorem of Arithmetic
  All integers can be expressed as a product of (powers of) primes
  Eg., 48 = 2*2*2*3 = 2^3 * 3
  Factorization is the process of finding the prime factors of a number.
    This is a relatively hard problem for large numbers.

Notation
  a|b is read "a divides b".  IE., a evenly divides b, or
  equivalently b*k=a for some integer k.

GCD - the greatest common divisor (or greatest common factor)
  Definition: The largest number which divides two numbers
  gcd(15, 25) = 5

Relatively prime numbers
  Two numbers whose gcd is 1.  (No common factors)

------- Modular math

Notation 
  % and (mod n).

Congruence
                         _                          _ 
  Equals with 3 bars.  a = b (mod n) means that a%n = b%n
  
The Modulus
  Simple exercies - multiplication, addition, congruence

More notation
  The fancy Z means the set of integers
  Zn means the set of integers modulo n; [0,n-1].
  Zn* is the multiplicative group of integers modulo n.  Quite
  different from Zn.

Multiplicative and additive inverses
  Additive inverses.  In Z, the additive inverse of 3 is -3, since 
  3 + -3 = 0, the additive identity.

  In Zn, the additive inverse of a is n-a, since a-(n-a) = n, which is
  congruent to 0 (mod n).

  In Z, the multiplicative inverse of 3 is 1/3, since 3*1/3=1.
  The multiplicative identity in both Z and Zn is 1.  The
  multiplicative inverse of 3 mod 10 is 7, since 7*3=21=1 (mod 10).
  This could be written 3^-1, or (rarely) 1/3.

Distribution in + and *
  Modular arithmetic is distributive.  That is, 

    a+b                   (mod n)
  = (a mod n) + (b mod n) (mod n)

  This distributive property should be intuitive when you think that 1
  o'clock pm plus 1 hour = 2pm, as does 1pm + 25 hours.  Mathematically,
  it works because:

  Let a=cn+d.  Then a%n=d, the remainder after taking out the
  multiples of n.

  Let b=en+f.  Then f = b%n.

     a    + b     (mod n)
   = cn+d + en+f  (mod n)

  but cn = en = 0    (mod n)   (since c and e are multiples of n), so:

  = d    + f     (mod n)
  = a%n  + b%n   (mod n).

  The modulus also distributes into multiplication.  Consider a*b%n.

  Let a=cn+d and b=en+f, just as before.

     a     * b                  (mod n)
  = (cn+d) * (en+f)             (mod n)
  = cnen   + cnf   + den   + df (mod n)
  = (cen)n + (cf)n + (de)n + df (mod n) 

  But any multiple of n modulo n is 0, so

  = 0      + 0     + 0     + df (mod n)
  = a%n * b%n                   (mod n)

  An example helps:

    7         * 26                   (mod 5)
  = (1*5 + 2) * (5*5 + 1)            (mod 5)
  = 1*5*5*5 + 1*5*1 + 2*5*5 + 2*1    (mod 5)
  = 0       + 0     + 0     + 2*1    (mod 5)
  = 7%5 + 26%5                       (mod 5)
  = 2                                (mod 5)

Modular exponentiation
  (See Lab 2 assignment on website for notes on modexp)

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Fermat's Little Theorem (and proof)

Using Fermat to compute multiplicative inverses

Fermat Primality test

Euler's Generalization of Fermat's Little Theorem

RSA:
Key generation
  * Safe primes
  * calculating phi(n) and using phi(phi(n)) to calculate ed=1 (mod phi(n))
Encryption and decryption
  * Easy once you have the keys
Signing
  * Using hashed documents

Groups, rings and fields.
  * Properties of each


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