Chapter 4 - Number Theory and Cryptography
4.1 - Divisibility and Modular Arithmetic
Definition 1 (Divisibility). Let a, b ∈ Z with a 6= 0. We say that a divides b (write a | b) iff there exists c ∈ Z such that b = ac. We write a - b if a does not divide b.
Theorem 1. Let a, b, c ∈ Z with a 6= 0.
1. If a | b and a | c then a | (b + c).
2. If a | b then a | bc.
3. If a | b and b | c then a | c.
4. If a | b and a | c then a | (mb + nc) for any m, n ∈ Z.
Theorem 2 (The Division Algorithm). Let a ∈ Z and d ∈ Z+. Then there exists unique q, r ∈ Z such that 0 ≤ r < d and a = dq + r. d is called the divisor, a is called the dividend, q is called the quotient, and r is called the remainder. The book uses the following notation for the quotient and remainder:
q = a div d, r = a mod d
Example 1.
1 4.3 - Primes and Greatest Common Divisors
Definition 2 (Prime). A prime is an integer such that if then either or .
Theorem 3 (Fundamental Theorem of Arithmetic). Every positive integer > 1 is either a prime or can be expressed as a product of primes. This product of primes is up to .
Definition 3. A composite integer is a positive integer > 1 which is not prime.
Example 2.
Proposition 1. An integer is divisible by 3 if and only if the sum of the digits in its decimal representation is divisible by 3. (This result is also true for 9.)
Example 3.
√ Proposition 2. If n is composite then n has a prime divisor less than or equal to n. Prove that 197 is prime:
2 Theorem 4. There are infinitely many primes.
Definition 4. Let a, b ∈ Z+, not both 0. d ∈ Z+ is said to be a common divisior of a and b iff d|a and d|b. d is the greatest common divisior of a and b iff
1. d is a common divisor of a and b, and 2. if e is any common divisor of a and b then e ≤ d.
We use the following notation to denote GCDs: d = gcd(a, b) or d = (a, b). (We’ll use the first one in this class to avoid confusion with points or open intervals.)
Example 4. a = 12, b = 8.
Definition 5. a, b ∈ Z are said to be relatively prime iff gcd(a, b) = 1. a1, a2, . . . , an are said to be pairwise relatively prime iff gcd(ai, aj) = 1 for any i 6= j.
Example 5. a = 12, b = 11, c = 35.
Definition 6. Let a, b ∈ Z. m ∈ Z is said to be a common multiple of a and b iff a|m and b|m. m is the least common multiple of a and b iff
1. m is a common multiple of a and b, and 2. if l is any common multiple of a and b then m ≤ l.
We use the following notation to denote LCMs: m = lcm(a, b) or m = [a, b]. (We’ll use the first one in this class to avoid confusion with closed intervals.)
Example 6. a = 12, b = 8.
3 Finding GCDs and LCMs using the prime factorization
Theorem 5. If a1 a2 an b1 b2 bn a = p1 p2 . . . pn and b = p1 p2 . . . pn where the pi are prime and ai, bj ∈ N then
min(a1,b1) min(a2,b2) min(an,bn) 1. gcd(a, b) = p1 p2 . . . pn
max(a1,b1) max(a2,b2) max(an,bn) 2. lcm(a, b) = p1 p2 . . . pn
Example 7. a = 61740 = 22 · 32 · 5 · 73, b = 1143450 = 2 · 33 · 52 · 7 · 112.
Theorem 6. a · b = gcd(a, b)lcm(a, b)
Theorem 7. Let a = bq + r. Then gcd(a, b) = gcd(b, r).
Theorem 8 (Bezout’s Theorem). Let a, b ∈ Z+. Then there exists u, v ∈ Z such that gcd(a, b) = au + bv. (We’ll refer to the u and v as the ‘Bezout’s coefficients’.)
Corollary 9. gcd(a, b) = 1 iff there exists u, v ∈ Z such that au + bv = 1.
4 The Extended Euclidean Algorithm (EEA)
We use the following table to compute gcd’s and find the Bezout’s coefficients. (This algorithm assumes a > b.)
i ri qi ui vi −1 a – 1 0 0 b – 0 1
1 r1 q1 u1 v1
2 r2 q2 u2 v2 ......
n rn qn un vn
n + 1 0 qn+1 – –
Table 1: EEA Table where ri−2 = ri−1qi + ri ⇐⇒ ri = ri−2 − ri−1qi for i ≥ 1 with 0 ≤ ri < ri−1 (so ri is the remainder when the division algorithm is applied to ri−2 and ri−1) and
ui = ui−2 − ui−1qi and vi = vi−2 − vi−1qi.
We stop when we get a remainder of 0, and the last non-zero remainder, rn, is the GCD of a and b.
At any line in the table ri = aui + bvi, so, in particular, un and vn are the Bezout’s coefficients.
Example 8. a = 1976 and b = 1251.
i ri qi ui vi
−1 1976 – 1 0
0 1251 – 0 1
1
2
3
4
5
6
7
8
9 0 – –
Table 2: Example 8
5 Example 9. a = 81 and b = 64. (There may be extra rows in the table I’ve given you.)
i ri qi ui vi
−1 81 – 1 0
0 64 – 0 1
1
2
3
4
5
6
Table 3: Example 9
6