Modular Arithmetic

(合同算術)

Discrete Mathematics I

13th lecture, Dec. 18, 2015

http://www.sw.it.aoyama.ac.jp/2015/Math1/lecture13.html

Martin J. Dürst

AGU

© 2005-15 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

 

Remaining Schedule

About makeup classes: The material in the makeup class is part of the final exam. If you have another makeup class at the same time, please inform the teacher as soon as possible.

補講について: 補講の内容は期末試験の対象。補講が別の補講とぶつかる場合には事前に申し出ること。

 

Questions about Final Exam

 

Summary of Algebraic Structures

(hierarchy of objects)

 

Applications of Bitwise Operations

Operations work on 8, 16, 32, or 64 bits concurrently

 

Other Bitwise Operations

 

Applications of Bitwise Operations

For many more advanced examples, see Hacker's Delight, Henry S. Warren, Jr., Addison-Wesley, 2003

 

Addition in Different Numeral Systems

Works the same as in decimal system:

Example (base 7):

Operand 1 3 6 5 1 2
Operand 2 6 0 3 3 4
carry 1 1 1
Result 1 3 0 1 4 6

 

Addition using Bitwise Operations

Single Digit Addition
0 1
0 0 1
1 1 10

 

Modular Arithmetic

 

Congruence Relation

 

Properties of Congruence Equations

 

Properties of the Modulo Operation

Reason: a(mod n) a mod n, and so on

 

Congruence and Groups

 

Congruence and Division

Division for rationals: a/b = ca b-1 = c; b b-1 = 1 (inverse)

Modular multiplicative inverse: bb-1 ≡ 1

n b 0 1 2 3 4 5 6 7
2 - 1
3 - 1 2
4 - 1 - 3
5 - 1 3 2 4
6 - 1 - - - 5
7 - 1 4 5 2 3 6
8 - 1 - 3 - 5 - 7

Only defined if n and b are coprime (i.e. GCD is 1)

Various methods to calculate

a/b (mod n) is defined as a b-1 (mod n)

Example: 3/4 (mod 7) = 3 · 4-1 (mod 7) = 3 · 2 (mod 7) = 6
(Check: 6 · 4 (mod 7) = 24 (mod 7) = 3)

 

Modulo Operation

See English Wikipedia article on Modulo Operation

 

An Example of Using the Modulo Operation

Output some data items, three items on a line.

A simple way:

int items = 0;
for (i=0; i<length; i++) {
    /* print out data[i] */
    items++;
    if (items==3) {
        printf("\n");
        items = 0;
    }
}

Using the modulo operation:

for (i=0; i<length; i++) {
    /* print out data[i] */
    if (i%3 == 2)  printf("\n");
}

Application of Congruence: Simple Calculation of Remainder

Example: 216 mod 29 = ?

216 = 25 · 25 · 25 · 2

216 = 25 · 25 · 25 · 2 = 32 · 32 · 32 · 2 ≡(mod 29) 3 · 3 · 3 · 2 = 54 ≡(mod 29) 25

 

Homework

Prepare for final exam

 

Glossary

rotation
回転
reflection
反射
hierarchy
階層
concurrent
同時
shift
シフト
invert
逆転、反転
modular arithmetic
合同算術
congruence (equation)
合同式
modulus
residue
剰余
modulo operation
剰余演算
congruence relation
合同関係
congruence class
合同類
residue class
剰余類
cyclic group
巡回群
modular multiplicative inverse
モジュラー逆数
operator
演算子
dividend
被除数
divisor
除数
implementation
実装