Modular Arithmetic

(合同算術)

Discrete Mathematics I

13th lecture, Jan. 9, 2015

http://www.sw.it.aoyama.ac.jp/2014/Math1/lecture13.html

Martin J. Dürst

© 2005-15 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Remaining schedule
• Summary and homework for last lecture
• Applications of bitwise operations
• Addition in different numeral systems
• Modular Arithmetic
• Modulo operation
• Digit sums and digital roots

Remaining Schedule

• January 16: 14th lecture
• January 23 (makeup class): 15th lecture
• January 30, 11:10～12:35: Final exam

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

Correction

In the definition of a Boolean algebra, the following was correct as printed:

• ∀a∈B: a▽0 = a, a△1 = a (0 is the neutral element for ▽, 1 is the neutral element for △)
• ∀a∈B: a▽¬a = 1, a△¬a = 0

(前回配布した資料のまま正しかった)

Last Week's Homework 1

If we define isomorphic groups as being "the same", there are two different groups of size 4. Give an example of each group as a Cayley table.

Hint: Check all the conditions (axioms) for a group. There will be a deduction if you use the same elements of the group as another student.

Solution 1: Cyclic group of order 4, interesting examples: Addition modulo 4, rotation by multiples of 90°, {1, 2, 3, 4} and multiplication modulo 5

[都合により削除]

Solution 2: Klein group, interesting examples: bitstrings of length 2 with XOR, reflections and rotations by 180°

[都合により削除]

Last Week's Homework 2

Draw the Hasse diagram of a Boolean algebra of order 4 (16 elements). There will be a deduction if you use the same elements of the group as another student.

Solution: For examples, see handouts of last week

Summary of Algebraic Structures

(hierarchy of objects)

Applications of Bitwise Operations

• Representation of sets and operations on sets
  (finite universal set, each bit position represents one element in the universal set)
• Storage and and check of properties
  (finite set of properties, each bit position represents one specific property)
• Detailled operations on data
  (example: data transformations, data compression)

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

Other Bitwise Operations

• Left shift:
  • b << n in C and many other programming languages
  • Shifts each of the bits in b by n positions to the left (more significant direction)
  • n bits its on the right are set to 0
  • The leftmost n bits in b disappear
  • If there is no overflow (disappearing 1 bits), b << n is equivalent to b ×2ⁿ
• Right shift:
  • b >> n in C and many other programming languages
  • Shifts each of the bits in b by n positions to the right (less significant direction)
  • n bits on the left are set to 0 or to the value of the leftmost bit in b,
    depending on programming lanugage and data type
  • The rightmost n bits in b disappear
  • b >> n is equivalent to b / 2ⁿ

Applications of Bitwise Operations

• In bitstring a, set bits from mask m:
  a | m
• In bitstring a, clear bits from mask m:
  a & ~m
• In bitstring a, invert bits from mask m:
  a ^ m
• In bitstring a, set bit number n:
  a | (1<<n) (rightmost bit is number 0)
• In bitstring a, clear bit number n:
  a & ~(1<<n) (rightmost bit is number 0)
• In bitstring a, invert bit number n:
  a ^ (1<<n) (rightmost bit is number 0)

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:

• Progress from least signifinant digit to more significant digits
• Carry over when base (e.g. 10) is reached

Example (base 7):

Addition using Bitwise Operations

• For inputs a=a₀ and b=b₀, calculate
  the digit-wise sum (without carry) as s₀ = a₀ ^ b₀, and
  the digit-wise carry as c₀ = a₀ & b₀
• Setting a₁ = s₀ and b₁ = c₀ << 1, repeat until c is 0
• Example: a₀ = 1011101、b₀ = 1101111

Modular Arithmetic

• Discovered and published in 1801 by Gauss
• Universal set is ℤ (integers)
• congruence (equation):

  a ≡_{(mod n)} b ⇔ a - b = qn ⇔ a = q₁n + r ∧ b = q₂n + r ∧ 0 ≦ r < n

• n is called modulus(n≠0)
• r is called residue
• a ≡_{(mod n)} b is also written a ≡ b (mod n) or a ≡ b (modulo n) or a ≡_n b

Congruence Relation

• Each modulus n creates a congruence relation on ℤ
• Congruence relations are equivalence relations
• The equivalence classes are called congruence classes or residue class
• The representatives k of the congruence classes are usually 0 ≦ k < n
• The representatives are the result of the modulo operation (Attention: depends on definition for negative numbers)

Properties of Congruence Equations

• a ≡ b ∧ c ≡ d ⇒ a + c ≡ b + d (mod n)
• a ≡ b ∧ c ≡ d ⇒ a - c ≡ b - d (mod n)
• a ≡ b ∧ c ≡ d ⇒ ac ≡ bd (mod n)
• a ≡ b ⇒ a^m ≡ b^m (mod n)
• a ≡ b ∧ c ≡ d ⇏ a / c ≡ b / d (mod n)

Properties of the Modulo Operation

• (a + c) mod n = (a mod n + c mod n) mod n (addition modulo n)
• (a - c) mod n = (a mod n - c mod n) mod n (substraction modulo n)
• (a · c) mod n = (a mod n · c mod n) mod n (multiplication modulo n)

Reason: a ≡_{(mod n)} a mod n, and so on

Congruence and Groups

• Addition modulo n creates a group on the congruence classes
• Multiplication modulo n creates a group of size n-1 on the congruence classes except 0 if n is prime

Congruence and Division

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

Modular multiplicative inverse): bb^-1 ≡ 1

Only defined if n are b 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

• Operator: % (C, Ruby and many other programming languages)、mod (Mathematics)
• Definitions for negative numbers:

  		remainder for negative operands
  operands		always non-negative		same sign as divisor		same sign as dividend
  a	n	q	r	q	r	q	r
  11	7	1	4	1	4	1	4
  -11	7	-2	3	-2	3	1	-4
  11	-7	-1	4	-2	-3	-1	4
  -11	-7	2	3	1	-4	-1	-4
  Programming languages,...		Pascal, this lecture		Ruby, Python, MS Excel		C (ISO 1999), Java, JavaScript, Perl, PHP

• In some programming languages (e.g. C before ISO 1999), the result is implementation-defined
• Some programming languages offer more than one function
• Always: qn+r=a ∧ |r|<|n|
• a ≡_n b ⇔ a mod n = b mod n only true for "always non-negative"

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++, items++) {
    /* print out data[i] */
    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

法

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

modular multiplicative inverse

モジュラー逆数

operator

演算子

dividend

被除数

divisor

除数

implementation

実装

digit sum

数字和

digital root

数字根