Number Theory
Tommy Färnqvist
Seminar 12
Number Theory
Outline
TDDD95: APS
Modular Arithmetic
Seminar in Algorithmic Problem Solving
April 29, 2016
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
Tommy Färnqvist
Department of Computer and Information Science
Linköping University
12.1
Outline
Number Theory
Tommy Färnqvist
Outline
1 Modular Arithmetic
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
2 Chinese Remainder Theorem
Final Note
3 Primes and Prime Testing
12.2
Modular Arithmetic (Zn )
Number Theory
Tommy Färnqvist
Definition
a ≡ b (modn) ⇔ n|(b − a), alternatively a = qn + b
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Zn for an integer n is an equivalence relation
Primes and Prime
Testing
Final Note
Definition (An equivalence class mod n)
[a] = {x | x ≡ a (modn)} = {a + qn | q ∈ Z}
• It is possible to perform arithmetic with equivalence classes mod n
• This is the topic of lab 3.5
12.3
Modular Inverse
Number Theory
Tommy Färnqvist
• What does it mean to calculate x/y mod n?
• Reformulate as x · y −1 mod n
• That is, we are looking for y −1 , such that y · y −1 mod n = 1 holds
Recall Euclid’s algorithm for greatest common divisor:
ull gcd(ull a, ull b) {
ull t;
while(b) t = a, a = b,b = t%b;
return a;
}
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
And the extended Euclidean algorithm, that finds x, y such that ax + by = gcd(a, b):
void exeuclid(ll a, ll b, ll *x ,ll *y) {
if(!b) *x = 1, *y = 0;
else exeuclid(b, a%b, y, x), *y -= *x * (a/b);
}
12.4
Chinese Remainder Theorem (labs 3.6-3.7)
Number Theory
Tommy Färnqvist
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
12.5
Chinese Remainder Theorem (labs 3.6-3.7)
Number Theory
Tommy Färnqvist
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
12.6
Primes
Number Theory
Tommy Färnqvist
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
12.7
Prime Testing (lab 3.8)
Number Theory
Tommy Färnqvist
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
12.8
Prime Generation
Number Theory
Tommy Färnqvist
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
12.9
Prime Testing and Generation
Number Theory
Tommy Färnqvist
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Primes and Prime
Testing
Final Note
12.10
Next time
Number Theory
Tommy Färnqvist
Outline
Modular Arithmetic
Chinese Remainder
Theorem
Computational Geometry. . .
Primes and Prime
Testing
Final Note
12.11