Modular Arithmetic

(合同算術)

Discrete Mathematics I

13th lecture, January 10, 2025

https://www.sw.it.aoyama.ac.jp/2024/Math1/lecture13.html

Martin J. Dürst

© 2005-25 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Remaining schedule
• Homework and summary of last lecture
• Applications of bitwise operations
• Modular arithmetic
• Properties of the modulo operation
• Congruence and division
• Digit sum and digital root

Remaining Schedule

• January 10: 13th lecture
• January 17: 14th lecture (makeup lecture/補講, same time, same place)
  Important: Contents of makeup class is part of final exam!
  If you have another makeup lecture at the same time, please tell me as soon as possible.
• January 24: 15th lecture
• January 31, 11:10-12:35 (85 min): Term Final Exams

How to use Past Exams

Important: Do not browse the problems and solutions!

1. Take time for a full exam (85 minutes)
2. Print out one exam
3. Solve this exam, with time limit
4. Look at the example solutions to check your solutions
5. Find out your weak points
6. Learn more in the areas you have identified as your weak points
7. Repeat starting at 1.

Important Points about Final Exam

• Most problems are in Japanese and English
• Most answers can be in Japanese or English
• Read problems carefully (distinguish between calculation, proof, explanation,...)
• Be able to explain concepts in your own words
• Be able to do calculations (base conversions, truth tables,...) speedily
• Combine and apply knowledge from different lectures
• Write clearly

Questions about Final Exam

Homework

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

Solution: For an example, see handout of last lecture

Caution:
A Hasse diagram does not have any horizontal lines!
A drawing of a four-dimensional cube is not a Hasse diagram if it has horizontal lines.

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 on 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

Modular Arithmetic

• Discovered and published in 1801 by Carl Fredrich 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	-1	4	-2	-3	-1	4
  -11	7	-2	3	-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

• Always true: qn+r=a ∧ |r|<|n|
• In some programming languages (e.g. C before ISO 1999), the result is implementation-defined
• Some programming languages offer more than one function
• 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 of size n 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

  (The group shown here, with n=5, is a cyclic group.)

Inverse and Division

Condition for inverse (b^-1) of b: b•b^-1 = b^-1•b = neutral element

Inverse for multiplication (multiplicative inverse): b b^-1 = b^-1 b = 1

Definition of division: a/b = a·1/b = a b^-1

Inverse for modular arithmetic (modular multiplicative inverse): b b^-1 ≡ b^-1 b ≡1 (mod n)

Modular Multiplicative Inverse

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

Various methods to calculate, very time-consuming

Modular Division

In general: a/b is defined as = a b^-1

For modular arithmetic: 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

Alternative: 3 · 3 · 3 · 2 = 27 · 2 ≡_{(mod 29)} -2 · 2 = -4 ≡_{(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

Digit Sum and Digital Root

Digit sum: Sum of all of the digits of a number

Digital root: Single-digit result of repeatedly calculating the digit sum (fixpoint)

Example in base 10:
The digit sum of 1839275 is 1+8+3+9+2+7+5 = 35
The digit sum of 35 is 3+5 = 8
The digital root of 1839275 is 8

Example in base 16:
The digit sum of A8FB is A+8+F+B (10+8+15+11) = (44) 2C
The digit sum of 2C is 2+C (2+12) = (14) E
The digital root of A8FB is E

Application of Congruence: Casting out Nines

• The digital root of a number n in base b is equal to n mod (b-1)
  (dr(n_b) = n mod (b-1))
• Reason: For n = d_k...d₁d₀ = d_k·b^k+ d _(k-1)·b ^(k-1)+...+d₁·b¹+d₀·b⁰,
  because b^m ≡_{(mod (b-1))} 1^m = 1, n ≡_{(mod (b-1))} d_k+ d_(k-1)+...+d₁+d₀
  
  
• If b=10, then b-1=9
• Example in base 10: 1839275₁₀ mod 9 = 8
• Example in base 16: A8FB₁₆ mod F = E
• This can be used for cross-checking the result of an arithmetic operation:
  x · y = z ⇒ dr(dr(x) · dr(y)) ≡_{mod (b-1)} dr(z)
  x · y ≠ z ⇐ dr(dr(x) · dr(y)) ≢_{mod (b-1)} dr(z)
  

Homework

(no need to submit)

• Standing Homework: Prepare for final exam using past exams.
• Create some addition problems in different bases, and then solve them (see lecture 2). Check your results using casting out nines (or b-1). Also, check your results by converting the numbers to decimal notation.
• Create some subtraction problems in different bases, and figure out how to do subtraction in bases other than 10. Check your results as above.
• Create some problems for remainder calculation using congruence, and solve them. Check your results using a calculator.
• Only one of the two equations below is correct. Which one?
  1346021 · 6292041 = 8469219318861
  3615137 · 8749019 = 31628902349603

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

digit

数字、桁

digit sum

数字和

digital root

数字根

casting out nines

九去法

implementation

実装

calculator

電卓