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Introduction to Number Theory

Lecture Outline

What we’ve covered so far:

– symmetric encryption

– hash functions

Where we are heading:

number theory

– public-key encryption

digital signatures

• Introduction to number theory

– divisibility

– GCD and Euclidean algorithm

– prime and composite numbers

– Chinese remainder theorem

– Euler Ø function

– Fermat’s theorem


• Divisibility

– given integers a and b, we say that b divides a (denoted by b|a) if b/a
is a whole number (or b = ac for an integer c)

– b is called a divisor of a

• Transitivity theorem

– we are given integers a, b, and c, where all of them > 1

– if a|b and b|c, then a|c

– proof

Linear combination theorem

– let a, b, c, x, and y be integers > 1

– if a|b and a|c, then a|(bx+cy)

• Proof

– a|b=>

– a|c =>

• Division algorithm (theorem)

– let a > 0 and b be two integers

– then there exist two unique integers q and r such that 0 ≤ r < a and
b = aq +r

to show existence:

uniqueness can be shown by contradiction:

• as sume there are integers q' and r' such that b = aq' +r' and
0 ≤ r' < a

• Notation

– the integer q is called the quotient

– the integer r is called the remainder

– [x] is the floor of x (largest integer ≤ x)

– [x] is the ceiling of x (smallest integer ≥ x)

– then q = [b/a] and r = b mod a

Greatest Greatest Divisor

• Greatest common divisor (GCD)

– suppose we are given integers a and b which are not both 0

– their greatest common divisor gcd(a, b) = c is the greatest number
that divides both a and b

– example: gcd(128, 100) = 4

– it is clear that gcd(a, b) = gcd(b, a)

GCD and multiplication

– we are given integers a, b, and m > 1

– if gcd(a,m) = gcd(b,m) = 1, then gcd(ab,m) = 1

– example: gcd(25, 7) = gcd(3, 7) = 1 => gcd(75, 7) = 1

• GCD and division

– Theorem 1

• we are given integers a and b

• if g = gcd(a, b), then

• example:

– Theorem 2

• if a is a positive integer and b, q, and r are integers with
b = aq +r, then gcd(b, a) = gcd(a, r)

• we can use this theorem to find GCD

Euclidean Algorithm

• Fact: given integers a > 0, b, q, and r such that b = aq +r,
gcd(a, b) = gcd(a, r)

• Euclidean algorithm for finding gcd(a, b)

– apply the division algorithm iteratively to compute the remainder

– the last non- zero remainder is the answer

– while a ≠ 0 do
r b mod a
b a
a r
return b

• Example:

– compute GCD of 165 and 285

steps of Euclidean algorithm:

– the answer is gcd(165, 285) =

Towards Extended Towards Extended Euclidean Algorithm

• Theorem:

– if integers a and b are not both 0, then there are integers x and y so
that ax +by = gcd(a, b)

– we can find x and y using the extended Euclidean algorithm

• Example:

– find x and y such that 285x +165y = gcd(285, 165) = 15

– we start with the next to last equation in our example and work

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