Properties of Arithmetic

1
Properties of Arithmetic

• Reference Pfleeger, Charles P., Security in
  Computing, 2nd Edition, Prentice Hall, 1996.

2
Identity and Inverses

• Let ? denote an operation on numbers.
• A number i is called an identity for operation
  ? if x ? i x and i ? x x for every number
  x.
• The number b is called the inverse of a under ?
  if a?bi.

3
Primes

• A prime number is any positive number that is
  divisible (with remainder 0) only by itself and
  1.
• A number that is not a prime is a composite.

4
Greatest Common Divisor

• The greatest common divisor of two integers, a
  and b, is the largest integer that evenly divides
  both a and b.
• Examples
• gcd(10,15) 5
• If p and q are primes, then gcd(p,q) 1.

5
Euclidean Algorithm

• Algorithm for finding the greatest common divisor
  , x of integers a and b.
• Suppose agtb.
• Then a mb r, where ma/b with remainder r.
• If xgcd(a,b) then x divides into a, mb, and r.

6
Euclidean Algorithm (p.2)

• But gcd(a,b) gcd(b,r).
• Then b mr r, where m b/r with remainder
  r.
• This process continues, until the remainder is 0.

7
Euclidean Algorithm (p.3)

• Example Find gcd(105,45).
• 105 245 15
• 45 315 0
• gcd(105,45) 15

8
Modular Arithmetic

• Modular arithmetic on nonnegative integers
  forms a commutative ring with operations addition
  and multiplication.
• If every number other than 0 has an inverse under
  multiplication, the group is a Galois field.
• The integers mod n are a Galois Field

9
Properties of Modular Arithmetic

• Associativity
• a (bc) mod n (ab)c mod n
• a (bc) mod n (ab)c mod n
• Commutativity
• ab mod n ba mod n
• ab mod n ba mod n

10
Properties of Modular Arithmetic (p.2)

• Distributivity
• a(bc) mod n ((ab) (ac)) mod n
• Existence of Identities
• a0 mod n 0a mod n a
• a1 mod n 1a mod n a

11
Properties of Modular Arithmetic (p.3)

• Existence of Inverses
• a (-a) mod n 0
• a(a-1) mod n 1 if a?0
• Reducibility
• (ab) mod n ((a mod n) (b mod n)) mod n
• (ab) mod n ((a mod n) (b mod n)) mod n

12
Fermats Theorem

• Let p be a prime and let a be an element such
  that altp.
• Then ap mod p a or ap-1 mod p 1.

13
Computing Inverses

• Let p be a prime and a lt p.
• Let x be the inverse of a.
• Then

ax mod p 1 (definition of inverse) ap-1 mod p
1 (Fermats Theorem) ax mod p ap-1 mod p x
mod p ap-2 mod p