Modular Arithmetic

(合同算術)

Discrete Mathematics I

13th lecture, January 10, 2020

https://www.sw.it.aoyama.ac.jp/2019/Math1/lecture13.html

Martin J. Dürst

© 2005-20 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Remaining schedule
• Homework and summary of last lecture
• Applications of bitwise operations
• Addition in different numeral systems
• Modular arithmetic
• Modulo operation
• Student Survey

Remaining Schedule

• January 10: this lecture
• January 17: 14th lecture (makeup class)
• January 24: 15th lecture
• January 31, 11:10-12:35: Term 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 now.

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今すぐ申し出ること。

Questions about Final Exam

Homework Due January 6

By using formula manipulation, show that the Wolfram axiom of Boolean logic (((A⊼B)⊼C)⊼(A⊼((A⊼C)⊼A))=C) is a tautology. For each simplification step, indicate which law(s) you used.

Hints: Simplify the left side to obtain the right side. There should be between 15 and 20 steps.

[Sorry, removed!]

Homework Due January 9

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 an example, see handout of last lecture

Summary of Last Lecture

• Boolean algebra is (an example of) an algebraic structure
• Boolean algebras have one set (B), two special elements (0 and 1), and three operations (⫬, △, and ▽)
• The axioms of boolean algebra are the same as the axioms of basic logic; there are various choices with tradeoffs between obviousness and compactness
• Boolean algebras are half-orders and lattices
• There are many examples of Boolean algebras: basic logic (with logic operations), bit strings (with bitwise logic operations), powersets (with set operations), certain sets of integers (with divisibility), gardens with flowers,...
• All finite Boolean algebras are of size 2ⁿ, and are isomorphic to an n-dimensional cube

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 check of (binary) properties
  (finite set of properties, each bit position represents one specific property)
  (examples: CPU features, character conversion UPPER↔lower case)
• Detailled operations on data
  (example: data transformations, data compression)

Operations work on 8, 16, 32, 64 or more 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 on the right are set to 0
  • The leftmost n bits in b disappear
  • If there is no overflow ('1' bits disappearing to the left), b << n is equivalent to b ×2ⁿ
  • Example: '101 1001' << 5 ⇒ '1011 0010 0000' (in actual program: 89 << 5 ⇒ 2848)
• 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 (integer division)
  • Example: '101 1001' >> 3 ⇒ '1011' (in actual program: 89 >> 3 ⇒ 11)

More 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 the 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=0

Example of Addition Using Bitwise Operations

Example: a₀ = 47 = 101111、b₀ = 58 = 111010

Modular Arithmetic

• Discovered and published in 1801 by Gauss
• Universal set is ℤ (integers)
• Congruence: Two integers are congruent modulo n if they have the same remainder when divided by n.
  (attention: for negative numbers, depends on definition of modulo operation)
• Congruence equation:

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

• 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 classes
• The representative k of a congruence class is usually choosen so that 0 ≦ k < n
• The representatives are the result of the modulo operation
  

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)
• But: 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

Where to use: a and c may be very large numbers, but a mod n and c mod n may be much smaller, so calculation becomes easier.

Modulo Operation for Negative Operands

• 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 (dividend)	n (divisor)	q (a/n)	r (a%n)	q (a/n)	r (a%n)	q (a/n)	r (a%n)
  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, Scheme, mathematics, this lecture		Ruby, Python, Perl, Excel		C (ISO 1999), Java, JavaScript, 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 true: 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, three items on a line.

A simple way:

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

Using the modulo operation:

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

Congruence and Groups

• Addition modulo n creates a group on the congruence classes (cyclic group)
• If n is prime, multiplication modulo n creates a group of size n-1 on the congruence classes except 0
  (used e.g. in Diffie-Hellman encryption)

  * mod 5	1	2	3	4
  1	1	2	3	4
  2	2	4	1	3
  3	3	1	4	2
  4	4	3	2	1

Congruence and Division

Division and inverse for rationals: a/b = c ⇔ a·1/b = a b^-1 = c

Condition for (multiplicative) inverse b^-1: b b^-1 = 1

Condition for modular multiplicative inverse: bb^-1 ≡ 1

The modular multiplicative inverse is only defined if n and b are coprime (i.e. GCD(n, b) = 1)

Various methods to calculate, very time-consuming

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)

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

Remainder Calculation: More Examples

3¹⁸ mod 7 = ?

3¹⁸ = (3³)⁶ = 27⁶ ≡_{(mod 7)} (-1)⁶ = 1

41¹⁰ mod 37 = ?

41¹⁰ ≡_{(mod 37)} 4¹⁰ = 2²⁰ = 32⁴ ≡_{(mod 37)} (-5)⁴ = 25² ≡_{(mod 37)} (-12)² = (6 · 2)² = 36 · 4 ≡_{(mod 37)} (-1) · 4 = -4 ≡_{(mod 37)} 33

Homework

Prepare for final exam

Student Survey

(授業改善のための学生アンケート)

WEB Survey

お願い: 自由記述に必ず良かった点、問題点を具体的に書きましょう

(悪い例: 発音が分かりにくい; 良い例: さ行が濁っているかどうか分かりにくい)

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

cyclic group

巡回群

modular multiplicative inverse

モジュラー逆数

coprime

互いに素

operator

演算子

dividend

被除数

divisor

除数

implementation

実装