Number Theory By, B. R. Chandavarkar CSE Dept., NITK Surathkal - - PowerPoint PPT Presentation

Basic Concepts in Number Theory

By,

• B. R. Chandavarkar

CSE Dept., NITK Surathkal

• Natural number: predecessor and successor. Example: 1, 2,

3, ….

• Whole number: natural number with 0.
• A factor of number is an exact divisor of that number.

– 1 is a factor of every number – Every number is a factor of itself – Every factor of a number is an exact divisor of that number – Every factor is less than or equal to the given number – Number of factors of a given number are finite – Every multiple of a number is greater than or equal to that number – The number of multiples of a given number is infinite – Every number is multiple of itself

• Perfect number: A number for which sum of all factors is

equal to twice the number is called perfect number. Example, 6 (1, 2, 3 and 6) and 28 (1, 2, 4, 7, 14 and 28)

• Prime number: The number other than 1 whose only

factors are 1 and number itself are called prime numbers.

• Composite number: Numbers having more than two

factors.

• Divisibility of numbers:

– Divisibility by 2: if a number has any of the digits 0, 2, 4, 6, or 8 in its

• ne place

– Divisibility by 3: if the sum of the digits is a multiple of 3 – Divisibility by 4: if the number formed by the last two digits is divisible by 4 – Divisibility by 5: if a number has 0 or 5 in its one position. – Divisibility by 6: if a number is divisible by 2 and 3 both – Divisibility by 8: if the number formed by the last three digits is divisible by 8. – Divisibility by 9: if the sum of the digits of a number is divisible by 9 – Divisibility by 10: if the number has 0 in the ones place

• Prime factorization: prime numbers are the factors.

Example, 24 (2 X 2 X 2 X 3), 980 (2 X 2 X 5 X 7 X 7)

• Co-prime Number: Two numbers having only 1 as a

common factor. Example: 4 & 15

• Additional Divisibility Rules

– If a number is divisible by another number than it is divisible by each

• f the factors of that number. Example: 24 divisible by 8 and also by

the factors of 8 i.e. 1, 2, 4 and 8. – If a number is divisible by two co-prime numbers than it is divisible by their product also. Example: 80 is divisible by 4 and 5 and also by 4 X 5 = 20. – If two given numbers are divisible by a number, then their sum is also divisible by that number. Example: 16 and 20 are both divisible by 5 and also 16 + 20. – If two given numbers are divisible by a number, then their difference is also divisible by that number. Example: 35 and 20 are both divisible by 5 and also 35 – 20.

• Highest Common Factor (HCF) OR Greatest Common

Divisor (GCD): HCF or GCD of two or more given numbers is the highest of their common factors.

– Example: (i) 20, 28 and 36 – 4

• Lowest Common Multiple (LCM): LCF of two or more

given numbers is the lowest of their common factors.

– Example: (i) 12 and 18 – 36 (ii) 24 and 90 (iii) 40, 48 and 45 (iv) 20, 25 and 30

• Integer: collection of whole numbers and negative numbers.
• Fraction: A fraction is a number representing part of a

whole.

• Improper

Fraction: numerator is bigger than denominator.

• Rational Number: Number that can be expressed in the

form p/q, where p and q are integers and q ≠ 0.

• Irrational Number: Number that cannot be expressed in

the form p/q, where p and q are integers and q ≠ 0.

• Real Number: Collection of rational and irrational

numbers.

Important Sets

• Definition 1.1.
• 1. A set is just a collection of elements. We usually denote a

set by enclosing its elements in braces “{ }”. [So, {1, 2, 3, 4} is a set whose elements are the numbers 1, 2, 3, and 4.]

• Sets don’t need to have numbers as elements, but they

likely will in this course.

• Note that the order that we write the elements of the

set does not matter, all that matters is the content, i.e., what elements it has.

• 2. An element of a set is said to belong to the set. We use

the symbol “---” for “belongs to” and “---” for “does not belong to”, such as in:

• 3. We have that

denotes the set of natural numbers. [The ellipsis here means “continues in the same way”.] Careful: Some authors exclude zero from the set of natural

• numbers. We will use instead

and refer as the set of positive integers. [Note that zero is neither positive nor negative!]

• 4. We define the set of integers as
• 5. We define the set of rational as

• A theorem is a statement [or proposition] whose validity

can be deduced from its assumptions by logical steps. So, it is something that you can deduce [the key word here] from

• ther facts.
• On the other hand, in mathematics often there is a hierarchy

for theorems:

– The term Theorem is reserved for statements that have greater

• importance. You probably know a few: Pythagoras’ Theorem,

Fundamental Theorem of Arithmetic, Thale’s Theorem, etc. – When a theorem is useful to us, but is of limited universal importance, the term Proposition is used. It is basically a “minor theorem”. – A Lemma is a theorem whose main purpose is to help prove one or more statements [which can be either full Theorems or mere Propositions]. – Finally, a Corollary is a result, which can be of some relative importance, but is an immediate [or almost immediate] consequence

• f a previous Theorem or Proposition.

• Note that we must always have a Proposition or Theorem,

and never a Corollary, following a Lemma. In the same way, a Corollary always comes after a Proposition or a Theorem, but never after a Lemma.

• We now review the concepts of greatest common divisor,

which we shall abbreviate by GCD, and least common multiple, which we shall abbreviate by LCM.

• The names already tell us what they mean: the GCD of two

integers a and b is the largest integer that divides a and b [at the same time], and the LCM is the smallest positive integer that is a multiple of a and of b [at the same time].

• We shall denote them gcd(a, b) and lcm(a, b) respectively.
• Note that for all positive integers a and b, we have that

gcd(a, b) ≥ 1 and lcm(a, b) ≤ ab.

• Moreover, since a divisor of a number is always less than or

equal to the number itself, and a multiple of a number is always greater than or equal to the number itself, we can also conclude that gcd(a, b) ≤ min(a, b) [where min(a, b) is the minimum between a and b] and lcm(a, b) ≥ max(a, b) [where max(a, b) is the maximum between a and b].

• In summary: 1 ≤ gcd(a, b) ≤ min(a, b) and max(a, b) ≤ lcm(a,

b) ≤ ab.

• GCD using Euclidean Algorithm
• LCM

Lemma: The product of two or more integers of the form 4n+1 is of the same form.

Congruence

Each residue class modulo n may be represented by any one

• f its members, although we usually represent each residue

class by the smallest nonnegative integer which belongs to that class (since this is the proper remainder which results from division). Any two members of different residue classes modulo n are incongruent modulo n. Furthermore, every integer belongs to one and only one residue class modulo n.

The set of integers {0, 1, 2, …, n − 1} is called the least residue system modulo n. Any set of n integers, no two of which are congruent modulo n, is called a complete residue system modulo n. It is clear that the least residue system is a complete residue system, and that a complete residue system is simply a set containing precisely one representative of each residue class modulo n. The least residue system modulo 4 is {0, 1, 2, 3}. Some other complete residue systems modulo 4 are: {1, 2, 3, 4}, {13, 14, 15, 16}, {−2, −1, 0, 1}, {−13, 4, 17, 18}, {−5, 0, 6, 21}, {27, 32, 37, 42} Some sets which are not complete residue systems modulo 4 are: {−5, 0, 6, 22} since 6 is congruent to 22 modulo 4. {5, 15} since a complete residue system modulo 4 must have exactly 4 incongruent residue classes.

Show that if a, b, and c are integers, then [a, b ]| c if and only if a | c and b | c .

Find the least positive residue of 2644 mod 645

Find all solutions of 9x is congruent to 12 (mod 15) Ans: 8, 13, and 18(3) Find all solutions of 7x congruent to 4 (12) Ans: -20 (4)

GALOIS FIELD – GROUP

• Group/ Albelian Group: A group G or {G, .} is a set of

elements with a binary operation denoted by . , that associates to each ordered pair (a, b) of elements in G an element (a . b) such that the following properties are obeyed:

– Closure: If a & b belong to G, then a . b also belongs to G. – Associative: For elements a, b & c in G, a . (b . c) = (a . b) . c. – Identity element: There is an element e in G such that a . e = e . a = a, for all a in G. – Inverse element: For each element a in G there is an element a’ in G such that a . a’ = a’ . a= e. – Commutative: for all elements a & b in G, a . b = b . a.

GALOIS FIELD – RING

• Ring/Commutative Ring: A ring R or {R, +, x} is a set of

elements with two binary operations , addition and multiplication, such that for all a, b & c in R the following properties are obeyed.

– All properties inside the definition of a ‘Group’ are obeyed. – Closure under multiplication: If a & b belong to R, then a x b also belongs to R. – Associativity of multiplication: a x (b x c) = (a x b) x c for all a, b & c in R. – Distributive laws: a x (b + c) = a x b + a x c; (a + b) x c = a x c + b x c for all a, b & c in R. – Commutativity of multiplication: a x b = b x a, for a & b in R. – Multiplicative identity: There is an element 1 in R such that a x 1 = 1 x a = a, for all a in R. – No zero divisors: If a, b in R and a x b = 0, then either a = 0 or b = 0.

GALOIS FIELD – FIELD

• Field: A field F or {F, +, x} is a set of elements with two

binary operations, addition and multiplication, such that for all a, b & c in F the following properties are obeyed.

– All properties inside the definition of ‘Group’ and ‘Ring’ are obeyed. – Multiplicative inverse: For each element a in F, except 0, there is an element a-1 in F such that aa-1 = (a-1)a = 1.

• Note: Finite field of the order pn, is written as GF (pn).
• We will study this field when n = 1 and when p = 2.
• Finite field of form GF (p): For a given prime p, finite

field of order p, GF (p), is defined as the set Zp of integers {0, 1, 2…..p-1} together with the arithmetic operations modulo p.

– Addition: a + b ↔ (a + b) mod p – Multiplication: a * b ↔ (a * b) mod p

• A finite field is also often known as a Galois field, after the French

mathematician Pierre Galois. A Galois field in which the elements can take q different values is referred to as GF(q). The formal properties of a finite field are: – (a) There are two defined operations, namely addition and multiplication. – (b) The result of adding or multiplying two elements from the field is always an element in the field. – (c) One element of the field is the element zero, such that a + 0 = a for any element a in the field. – (d) One element of the field is unity, such that a • 1 = a for any element a in the field. – (e) For every element a in the field, there is an additive inverse element -a, such that a + ( - a) = 0. This allows the operation of subtraction to be defined as addition of the inverse. – (f) For every non-zero element b in the field there is a multiplicative inverse element b-1 such that b b-1= 1. This allows the operation of division to be defined as multiplication by the inverse. – (g) The associative [a + (b + c) = (a + b) + c, a • (b • c) = [(a • b) • c], commutative [a + b = b + a, a • b = b • a], and distributive [a • (b + c) = a • b + a • c] laws apply.

GALOIS FIELD OF FORM GF(P)

Construct addition and multiplication tables over GF(7), then show how you can make subtraction and division operations over this field.

Group, Rings and Fields

• Groups, rings, and fields are the fundamental elements of a

branch of mathematics known as abstract algebra, or modern algebra.

• In abstract algebra, we are concerned with sets on whose

elements we can operate algebraically; that is, we can combine two elements of the set, perhaps in several ways, to obtain a third element of the set.

Finite Field of the form GF(p)

• Infinite fields are not of particular interest in the context of
• cryptography. However, finite fields play a crucial role in

many cryptographic algorithms. It can be shown that the

• rder of a finite field (number of elements in the field) must

be a power of a prime pn, where n

• The finite field of order pn is generally written GF(pn); GF

stands for Galois field, in honour of the mathematician who first studied finite fields.

• Two special cases are of interest for our purposes. For n = 1,

we have the finite field GF(p); this finite field has a different structure than that for finite fields with n > 1.

• We look at finite fields of the form GF(2n).

Finite Fields of Order p

• For a given prime, p, we define the finite field of order p,

GF(p), as the set Zp of integers {0, 1, c, p - 1} together with the arithmetic operations modulo p.

• The set Zn of integers {0, 1, c, n - 1}, together with the

arithmetic operations modulo n, is a commutative ring (Table 4.3).

• We further observed that any integer in Zn has a

multiplicative inverse if and only if that integer is relatively prime to n.

• If n is prime, then all of the nonzero integers in Zn are

relatively prime to n, and therefore there exists a multiplicative inverse for all of the nonzero integers in Zn.

• Thus, for Zp we can add the following properties to those

listed in Table 4.3:

• Multiplicative inverse (w-1) - For each w Є Zp, w ≠ 0, there

exists a z Є Zp such that w * z is congruent to (mod p)

• Example: a = 1759, b = 550 then 1759x + 550y = d. The

value of y = 355. To verify 550 X 355 mod 1759 = 1

Computations with Polynomials

• Next we consider computations with polynomials whose

coefficients are from the binary field GF(2). f(x) = f0 + f1x + f2x2 + ….. + fnxn

• The degree of a polynomial is the largest power of X with a

nonzero coefficient.

• Example: There are two polynomials over GF(2) with degree

1: X and 1+X.

• Example: There are four polynomials over GF(2) with degree

2: X2, 1 + X2, X + X2, and 1 + X + X2.

• In general, there are 2n polynomials over GF(2) with degree
• n. Polynomials over GF(2) can be added (or subtracted),

multiplied, and divided in the usual way.

• Add a(X)=1+X+X3 + X5 with b(X)=1+ X2+ X3+ X4+ X7 ?

• Suppose that the degree of g(X) is not zero. When f(X) is

divided by g(X), we obtain a unique pair of polynomials over GF(2)—q(X), called the quotient, and r(X), called the remainder—such that f(x) = g(x) q(x) + r(x)

• Example: f(x) = 1 + x + x4 + x5 + x6 and g(x) = 1 + x + x3

results into q(x) = x3 + x2 and r(x) = x2 + x + 1

• When f (X) is divided by g(X), if the remainder r(X) is

identical to zero [r(X) = 0], we say that f(X) is divisible by g(X) and g(X) is a factor of f(X).

• For real numbers, if a is a root of a polynomial f(X) [i.e., f(a)

= 0], f(X) is divisible by x — a.

• For a polynomial f(X) over GF(2), if it has an even number of

terms, it is divisible by X+ 1.

• A polynomial p(X) over GF(2) of degree m is said to be

irreducible over GF(2) if p(X) is not devisable by any polynomial over GF(2) of degree less than m but greater than zero.