cryptography 2 is a easy ppt of computer science

Cryptography and
Network Security
Eighth Edition, Global Edition
by William Stallings
Copyright © 2023 Pearson Education, Ltd. All Rights Reserved..

Chapter 5
Finite Fields

Groups
• 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 ) in G , such that the following axioms are obeyed:
• (A1) Closure:
• If a and b belong to G, then a  b is also in G
• (A2) Associative:
• a  (b  c) = (a  b)  c for all a, b, c in G
• (A3) Identity element:
• There is an element e in G such that a  e = e  a = a for all a in G
• (A4) Inverse element:
• For each a in G, there is an element a1
in G such that aa1 = a1  a = e
• (A5) Commutative:
• a  b = b  a for all a, b in G

Cyclic Group
• Exponentiation is defined within a group as a repeated
application of the group operator, so that a3
= aaa
• We define a0
= e as the identity element, and a-
n
= (a’
)n
, where a’
is the inverse element of a within the
group
• A group G is cyclic if every element of G is a power ak
(k is an integer) of a fixed element a € G
• The element a is said to generate the group G or to be
a generator of G
• A cyclic group is always abelian and may be finite or
infinite

Rings
• A ring R , sometimes denoted by {R , + , * }, is a set of elements with two binary
operations, called addition and multiplication, such that for all a , b , c in R the following
axioms are obeyed:
(A1–A5)
R is an abelian group with respect to addition; that is, R satisfies axioms A1 through A5. For
the case of an additive group, we denote the identity element as 0 and the inverse of a as –
a
(M1) Closure under multiplication:
If a and b belong to R , then ab is also in R
(M2) Associativity of multiplication:
a (bc ) = (ab)c for all a , b , c in R
(M3) Distributive laws:
a (b + c ) = ab + ac for all a , b , c in R
(a + b )c = ac + bc for all a , b , c in R
• In essence, a ring is a set in which we can do addition, subtraction [a - b = a + (-b )], and
multiplication without leaving the set

Rings (cont.)
• A ring is said to be commutative if it satisfies the
following additional condition:
(M4) Commutativity of multiplication:
ab = ba for all a, b in R
• An integral domain is a commutative ring that obeys
the following axioms.
(M5) Multiplicative identity:
There is an element 1 in R such that a1 = 1a = a
for all a in R
(M6) No zero divisors:
If a , b in R and ab = 0, then either a = 0 or b = 0

Fields
• A field F , sometimes denoted by {F, +,* }, is a set of elements with
two binary operations, called addition and multiplication, such that
for all a, b, c in F the following axioms are obeyed:
(A1–M6)
F is an integral domain; that is, F satisfies axioms A1 through A5 and M1
through M6
(M7) Multiplicative inverse:
For each a in F, except 0, there is an element a-1
in F such that aa-1
= (a-1
)a = 1
• In essence, a field is a set in which we can do addition, subtraction,
multiplication, and division without leaving the set. Division is
defined with the following rule: a /b = a (b-1
)
Familiar examples of fields are the rational numbers, the real
numbers, and the complex numbers. Note that the set of all integers is
not a field, because not every element of the set has a multiplicative
inverse.

Finite Fields of the Form
GF(p)
• Finite fields play a crucial role in many
cryptographic algorithms
• It can be shown that the order of a finite
field must be a power of a prime pn
, where
n is a positive integer
• The finite field of order pn
is generally
written GF(pn
)
• GF stands for Galois field, in honor of the
mathematician who first studied finite
fields

Table 5.1(a)
(a) Addition modulo 8

Table 5.1(b)
(b) Multiplication modulo 8

Table 5.1(c)
(c) Additive and multiplicative
inverses modulo 8

Table 5.1(d)
(d) Addition modulo 7

Table 5.1(e)
(e) Multiplication modulo 7

Table 5.1(f)
(f) Additive and multiplicative
inverses modulo 7

we have shown
how to
construct a
finite field of
order p, where
p is prime.
GF(p) is defined
with the
following
properties:
• 1. GF(p) consists of p
elements
• 2. The binary operations +
and * are defined over the
set. The operations of
addition, subtraction,
multiplication, and division
can be performed without
leaving the set. Each
element of the set other
than 0 has a multiplicative
inverse
• We have shown that the
elements of GF(p) are the
integers {0, 1, . . . , p – 1} and
that the arithmetic
operations are addition and
multiplication mod p

Polynomial Arithmetic
With Coefficients in Zp
• If each distinct polynomial is considered to be an
element of the set, then that set is a ring
• When polynomial arithmetic is performed on
polynomials over a field, then division is possible
• Note: this does not mean that exact division is possible
• If we attempt to perform polynomial division over a
coefficient set that is not a field, we find that division is
not always defined
• Even if the coefficient set is a field, polynomial division is
not necessarily exact
• With the understanding that remainders are allowed, we
can say that polynomial division is possible if the coefficient
set is a field

Polynomial Division
• We can write any polynomial in the form:
f(x) = q(x) g(x) + r(x)
• r(x) can be interpreted as being a remainder
• So r(x) = f(x) mod g(x)
• If there is no remainder we can say g(x) divides f(x)
• Written as g(x) | f(x)
• We can say that g(x) is a factor of f(x)
• Or g(x) is a divisor of f(x)
• A polynomial f(x) over a field F is called irreducible if and
only if f(x) cannot be expressed as a product of two
polynomials, both over F, and both of degree lower than
that of f(x)
• An irreducible polynomial is also called a prime polynomial

Example of Polynomial Arithmetic
Over GF(2)
(Figure 5.6 can be found on page 129 in the textb

Polynomial GCD
• The polynomial c(x) is said to be the greatest common
divisor of a(x) and b(x) if the following are true:
• c(x) divides both a(x) and b(x)
• Any divisor of a(x) and b(x) is a divisor of c(x)
• An equivalent definition is:
• gcd[a(x), b(x)] is the polynomial of maximum degree
that divides both a(x) and b(x)
• The Euclidean algorithm can be extended to find the
greatest common divisor of two polynomials whose
coefficients are elements of a field

Table 5.2(a)
Arithmetic in GF(23
)

Table 5.2(b)
Arithmetic in GF(23
)

Table 5.2(c)
Arithmetic
in GF(23
)
(c) Additive and multiplicative inverses

Table 5.3
Polynomial Arithmetic Modulo (x3
+ x + 1)
(a) Addition
(Table is on page 136 in the textbook)

Table 5.3
Polynomial Arithmetic Modulo (x3
+ x + 1)
(b) Multiplication

Table 5.4
Extended Euclid [(x8
+ x4
+ x3
+ x + 1), (x7
+ x + 1)]
(Table 5.4 can be found on page 138 in textbook)

Computational
Considerations
• Since coefficients are 0 or 1, they can
represent any such polynomial as a bit
string
• Addition becomes XOR of these bit strings
• Multiplication is shift and XOR
• cf long-hand multiplication
• Modulo reduction is done by repeatedly
substituting highest power with
remainder of irreducible polynomial (also

Using a Generator
• A generator g of a finite field F of order q
(contains q elements) is an element whose first
q-1 powers generate all the nonzero elements of
F
• The elements of F consist of 0, g0
, g1
, . . . ., gq-2
• Consider a field F defined by a polynomial fx
• An element b contained in F is called a root of the
polynomial if f(b) = 0
• Finally, it can be shown that a root g of an
irreducible polynomial is a generator of the
finite field defined on that polynomial

Table 5.5
Generator for GF(23
) using x3
+ x + 1

Table 5.6
GF(23
) Arithmetic Using Generator for the Polynomial (x3
+
x + 1)
(a) Addition

Table 5.6
GF(23
) Arithmetic Using Generator for the Polynomial (x3
+
x + 1)
(b) Multiplication

Summary
• Distinguish among
groups, rings, and
fields
• Define finite fields
of the form GF(p)
• Define finite fields
of the form GF(2n
)
• Explain the
differences among
ordinary polynomial
arithmetic,
polynomial
arithmetic with
coefficients in Zp,
and modular
polynomial
arithmetic in GF(2n
)
• Explain the two
different uses of the
mod operator

cryptography 2 is a easy ppt of computer science

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