ch04 cryptography1cryptography1cryptography1

Cryptography and
Cryptography and
Network Security
Network Security
Chapter 4
Chapter 4
Fourth Edition
Fourth Edition
by William Stallings
by William Stallings
Lecture slides by Lawrie Brown
Lecture slides by Lawrie Brown

Chapter 4 – Finite Fields
Chapter 4 – Finite Fields
The next morning at daybreak, Star flew indoors,
The next morning at daybreak, Star flew indoors,
seemingly keen for a lesson. I said, "Tap eight." She did
seemingly keen for a lesson. I said, "Tap eight." She did
a brilliant exhibition, first tapping it in 4, 4, then giving me
a brilliant exhibition, first tapping it in 4, 4, then giving me
a hasty glance and doing it in 2, 2, 2, 2, before coming
a hasty glance and doing it in 2, 2, 2, 2, before coming
for her nut. It is astonishing that Star learned to count up
for her nut. It is astonishing that Star learned to count up
to 8 with no difficulty, and of her own accord discovered
to 8 with no difficulty, and of her own accord discovered
that each number could be given with various different
that each number could be given with various different
divisions, this leaving no doubt that she was consciously
divisions, this leaving no doubt that she was consciously
thinking each number. In fact, she did mental arithmetic,
thinking each number. In fact, she did mental arithmetic,
although unable, like humans, to name the numbers. But
although unable, like humans, to name the numbers. But
she learned to recognize their spoken names almost
she learned to recognize their spoken names almost
immediately and was able to remember the sounds of
immediately and was able to remember the sounds of
the names. Star is unique as a wild bird, who of her own
the names. Star is unique as a wild bird, who of her own
free will pursued the science of numbers with keen
free will pursued the science of numbers with keen
interest and astonishing intelligence.
interest and astonishing intelligence.
—
— Living with Birds
Living with Birds, Len Howard
, Len Howard

Introduction
Introduction
 will now introduce finite fields
will now introduce finite fields
 of increasing importance in cryptography
of increasing importance in cryptography

AES, Elliptic Curve, IDEA, Public Key
AES, Elliptic Curve, IDEA, Public Key
 concern operations on “numbers”
concern operations on “numbers”

where what constitutes a “number” and the
where what constitutes a “number” and the
type of operations varies considerably
type of operations varies considerably
 start with concepts of groups, rings, fields
start with concepts of groups, rings, fields
from abstract algebra
from abstract algebra

Group
Group
 a set of elements or “numbers”
a set of elements or “numbers”
 with some operation whose result is also in
with some operation whose result is also in
the set (closure)
the set (closure)
 obeys:
obeys:

associative law:
associative law: (a.b).c = a.(b.c)
(a.b).c = a.(b.c)

has identity
has identity e
e:
: e.a = a.e = a
e.a = a.e = a

has inverses
has inverses a
a-1
-1
:
: a.a
a.a-1
-1
= e
= e
 if commutative
if commutative a.b = b.a
a.b = b.a

then forms an
then forms an abelian group
abelian group

Cyclic Group
Cyclic Group
 define
define exponentiation
exponentiation as repeated
as repeated
application of operator
application of operator

example:
example: a
a-3
-3
= a.a.a
= a.a.a
 and let identity be:
and let identity be: e=
e=a
a0
0
 a group is cyclic if every element is a
a group is cyclic if every element is a
power of some fixed element
power of some fixed element

ie
ie b =
b = a
ak
k
for some
for some a
a and every
and every b
b in group
in group
 a
a is said to be a generator of the group
is said to be a generator of the group

Ring
Ring
 a set of “numbers”
a set of “numbers”
 with two operations (addition and multiplication)
with two operations (addition and multiplication)
which form:
which form:
 an abelian group with addition operation
an abelian group with addition operation
 and multiplication:
and multiplication:

has closure
has closure

is associative
is associative

distributive over addition:
distributive over addition: a(b+c) = ab + ac
a(b+c) = ab + ac
 if multiplication operation is commutative, it forms a
if multiplication operation is commutative, it forms a
commutative ring
commutative ring
 if
if multiplication operation has an identity and no
multiplication operation has an identity and no
zero divisors, it forms an
zero divisors, it forms an integral domain
integral domain

Field
Field
 a set of numbers
a set of numbers
 with two operations which form:
with two operations which form:

abelian group for addition
abelian group for addition

abelian group for multiplication (ignoring 0)
abelian group for multiplication (ignoring 0)

ring
ring
 have hierarchy with more axioms/laws
have hierarchy with more axioms/laws

group -> ring -> field
group -> ring -> field

Modular Arithmetic
Modular Arithmetic
 define
define modulo operator
modulo operator “
“a mod n”
a mod n” to be
to be
remainder when a is divided by n
remainder when a is divided by n
 use the term
use the term congruence
congruence for:
for: a = b mod n
a = b mod n

when divided by
when divided by n,
n, a & b have same remainder
a & b have same remainder

eg. 100 = 34 mod 11
eg. 100 = 34 mod 11
 b is called a
b is called a residue
residue of a mod n
of a mod n

since with integers can always write:
since with integers can always write: a = qn + b
a = qn + b

usually chose smallest positive remainder as residue
usually chose smallest positive remainder as residue
• ie.
ie. 0 <= b <= n-1
0 <= b <= n-1

process is known as
process is known as modulo reduction
modulo reduction
• eg. -12 mod 7
eg. -12 mod 7 =
= -5 mod 7
-5 mod 7 =
= 2 mod 7
2 mod 7 =
= 9 mod 7
9 mod 7

Divisors
Divisors
 say a non-zero number
say a non-zero number b
b divides
divides a
a if for
if for
some
some m
m have
have a=mb
a=mb (
(a,b,m
a,b,m all integers)
all integers)
 that is
that is b
b divides into
divides into a
a with no remainder
with no remainder
 denote this
denote this b|a
b|a
 and say that
and say that b
b is a
is a divisor
divisor of
of a
a
 eg. all of 1,2,3,4,6,8,12,24 divide 24
eg. all of 1,2,3,4,6,8,12,24 divide 24

Modular Arithmetic Operations
Modular Arithmetic Operations
 is 'clock arithmetic'
is 'clock arithmetic'
 uses a finite number of values, and loops
uses a finite number of values, and loops
back from either end
back from either end
 modular arithmetic is when do addition &
modular arithmetic is when do addition &
multiplication and modulo reduce answer
multiplication and modulo reduce answer
 can do reduction at any point, ie
can do reduction at any point, ie

a+b mod n = [a mod n + b mod n] mod n
a+b mod n = [a mod n + b mod n] mod n

Modular Arithmetic
Modular Arithmetic
 can do modular arithmetic with any group of
can do modular arithmetic with any group of
integers:
integers: Z
Zn
n = {0, 1, … , n-1}
= {0, 1, … , n-1}
 form a commutative ring for addition
form a commutative ring for addition
 with a multiplicative identity
with a multiplicative identity
 note some peculiarities
note some peculiarities

if
if (a+b)
(a+b)=(a+c) mod n
=(a+c) mod n
then
then b=c mod n
b=c mod n

but if
but if (a.b)
(a.b)=(a.c) mod n
=(a.c) mod n
then
then b=c mod n
b=c mod n only if
only if a
a is relatively prime to
is relatively prime to n
n

Modulo 8 Addition Example
Modulo 8 Addition Example
+ 0 1 2 3 4 5 6 7
0 0 1 2 3 4 5 6 7
1 1 2 3 4 5 6 7 0
2 2 3 4 5 6 7 0 1
3 3 4 5 6 7 0 1 2
4 4 5 6 7 0 1 2 3
5 5 6 7 0 1 2 3 4
6 6 7 0 1 2 3 4 5
7 7 0 1 2 3 4 5 6

Greatest Common Divisor (GCD)
Greatest Common Divisor (GCD)
 a common problem in number theory
a common problem in number theory
 GCD (a,b) of a and b is the largest number
GCD (a,b) of a and b is the largest number
that divides evenly into both a and b
that divides evenly into both a and b

eg GCD(60,24) = 12
eg GCD(60,24) = 12
 often want
often want no common factors
no common factors (except 1)
(except 1)
and hence numbers are
and hence numbers are relatively prime
relatively prime

eg GCD(8,15) = 1
eg GCD(8,15) = 1

hence 8 & 15 are relatively prime
hence 8 & 15 are relatively prime

Euclidean Algorithm
Euclidean Algorithm
 an efficient way to find the GCD(a,b)
an efficient way to find the GCD(a,b)
 uses theorem that:
uses theorem that:

GCD(a,b) = GCD(b, a mod b)
GCD(a,b) = GCD(b, a mod b)
 Euclidean Algorithm to compute GCD(a,b) is:
Euclidean Algorithm to compute GCD(a,b) is:
EUCLID(a,b)
EUCLID(a,b)
1. A
1. A =
= a; B
a; B =
= b
b
2. if B = 0 return A = gcd(a, b)
2. if B = 0 return A = gcd(a, b)
3. R = A mod B
3. R = A mod B
4. A = B
4. A = B
5. B
5. B =
= R
R
6. goto 2
6. goto 2

Example GCD(1970,1066)
Example GCD(1970,1066)
1970 = 1 x 1066 + 904
1970 = 1 x 1066 + 904 gcd(1066, 904)
gcd(1066, 904)
1066 = 1 x 904 + 162
1066 = 1 x 904 + 162 gcd(904, 162)
gcd(904, 162)
904 = 5 x 162 + 94
904 = 5 x 162 + 94 gcd(162, 94)
gcd(162, 94)
162 = 1 x 94 + 68
162 = 1 x 94 + 68 gcd(94, 68)
gcd(94, 68)
94 = 1 x 68 + 26
94 = 1 x 68 + 26 gcd(68, 26)
gcd(68, 26)
68 = 2 x 26 + 16
68 = 2 x 26 + 16 gcd(26, 16)
gcd(26, 16)
26 = 1 x 16 + 10
26 = 1 x 16 + 10 gcd(16, 10)
gcd(16, 10)
16 = 1 x 10 + 6
16 = 1 x 10 + 6 gcd(10, 6)
gcd(10, 6)
10 = 1 x 6 + 4
10 = 1 x 6 + 4 gcd(6, 4)
gcd(6, 4)
6 = 1 x 4 + 2
6 = 1 x 4 + 2 gcd(4, 2)
gcd(4, 2)
4 = 2 x 2 + 0
4 = 2 x 2 + 0 gcd(2, 0)
gcd(2, 0)

Galois Fields
Galois Fields
 finite fields play a key role in cryptography
finite fields play a key role in cryptography
 can show number of elements in a finite
can show number of elements in a finite
field
field must
must be a power of a prime p
be a power of a prime pn
n
 known as Galois fields
known as Galois fields
 denoted GF(p
denoted GF(pn
n
)
)
 in particular often use the fields:
in particular often use the fields:

GF(p)
GF(p)

GF(2
GF(2n
n
)
)

Galois Fields GF(p)
Galois Fields GF(p)
 GF(p) is the set of integers {0,1, … , p-1}
GF(p) is the set of integers {0,1, … , p-1}
with arithmetic operations modulo prime p
with arithmetic operations modulo prime p
 these form a finite field
these form a finite field

since have multiplicative inverses
since have multiplicative inverses
 hence arithmetic is “well-behaved” and
hence arithmetic is “well-behaved” and
can do addition, subtraction, multiplication,
can do addition, subtraction, multiplication,
and division without leaving the field GF(p)
and division without leaving the field GF(p)

GF(7) Multiplication Example
GF(7) Multiplication Example
 0 1 2 3 4 5 6
0 0 0 0 0 0 0 0
1 0 1 2 3 4 5 6
2 0 2 4 6 1 3 5
3 0 3 6 2 5 1 4
4 0 4 1 5 2 6 3
5 0 5 3 1 6 4 2
6 0 6 5 4 3 2 1

Finding Inverses
Finding Inverses
EXTENDED EUCLID(
EXTENDED EUCLID(m
m,
, b
b)
)
1.
1. (A1, A2, A3)=(1, 0,
(A1, A2, A3)=(1, 0, m
m);
);
(B1, B2, B3)=(0, 1,
(B1, B2, B3)=(0, 1, b
b)
)
2. if
2. if B3 = 0
B3 = 0
return
return A3 = gcd(
A3 = gcd(m
m,
, b
b); no inverse
); no inverse
3. if
3. if B3 = 1
B3 = 1
return
return B3 = gcd(
B3 = gcd(m
m,
, b
b); B2 =
); B2 = b
b–1
–1
mod
mod m
m
4.
4. Q = A3 div B3
Q = A3 div B3
5.
5. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)
(T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)
6.
6. (A1, A2, A3)=(B1, B2, B3)
(A1, A2, A3)=(B1, B2, B3)
7.
7. (B1, B2, B3)=(T1, T2, T3)
(B1, B2, B3)=(T1, T2, T3)
8. goto
8. goto 2
2

Inverse of 550 in GF(1759)
Inverse of 550 in GF(1759)
Q A1 A2 A3 B1 B2 B3
— 1 0 1759 0 1 550
3 0 1 550 1 –3 109
5 1 –3 109 –5 16 5
21 –5 16 5 106 –339 4
1 106 –339 4 –111 355 1

Polynomial Arithmetic
Polynomial Arithmetic
 can compute using polynomials
can compute using polynomials
f
f(
(x
x) = a
) = an
nx
xn
n
+ a
+ an-1
n-1x
xn-1
n-1
+ … + a
+ … + a1
1x +
x + a
a0
0 = ∑ a
= ∑ ai
ix
xi
i
• nb. not interested in any specific value of x
nb. not interested in any specific value of x
• which is known as the indeterminate
which is known as the indeterminate
 several alternatives available
several alternatives available

ordinary polynomial arithmetic
ordinary polynomial arithmetic

poly arithmetic with coords mod p
poly arithmetic with coords mod p

poly arithmetic with coords mod p and
poly arithmetic with coords mod p and
polynomials mod m(x)
polynomials mod m(x)

Ordinary Polynomial Arithmetic
Ordinary Polynomial Arithmetic
 add or subtract corresponding coefficients
add or subtract corresponding coefficients
 multiply all terms by each other
multiply all terms by each other
 eg
eg
let
let f
f(
(x
x) =
) = x
x3
3
+
+ x
x2
2
+ 2 and
+ 2 and g
g(
(x
x) =
) = x
x2
2
–
– x
x + 1
+ 1
f
f(
(x
x) +
) + g
g(
(x
x) =
) = x
x3
3
+ 2
+ 2x
x2
2
–
– x
x + 3
+ 3
f
f(
(x
x) –
) – g
g(
(x
x) =
) = x
x3
3
+
+ x
x + 1
+ 1
f
f(
(x
x) x
) x g
g(
(x
x) =
) = x
x5
5
+ 3
+ 3x
x2
2
– 2
– 2x
x + 2
+ 2

Polynomial Arithmetic with
Polynomial Arithmetic with
Modulo Coefficients
Modulo Coefficients
 when computing value of each coefficient
when computing value of each coefficient
do calculation modulo some value
do calculation modulo some value

forms a polynomial ring
forms a polynomial ring
 could be modulo any prime
could be modulo any prime
 but we are most interested in mod 2
but we are most interested in mod 2

ie all coefficients are 0 or 1
ie all coefficients are 0 or 1

eg. let
eg. let f
f(
(x
x) =
) = x
x3
3
+
+ x
x2
2
and
and g
g(
(x
x) =
) = x
x2
2
+
+ x
x + 1
+ 1
f
f(
(x
x) +
) + g
g(
(x
x) =
) = x
x3
3
+
+ x
x + 1
+ 1
f
f(
(x
x) x
) x g
g(
(x
x) =
) = x
x5
5
+
+ x
x2
2

Polynomial Division
Polynomial Division
 can write any polynomial in the form:
can write any polynomial in the form:

f
f(
(x
x) =
) = q
q(
(x
x)
) g
g(
(x
x) +
) + r
r(
(x
x)
)

can interpret
can interpret r
r(
(x
x)
) as being a remainder
as being a remainder

r
r(
(x
x) =
) = f
f(
(x
x) mod
) mod g
g(
(x
x)
)
 if have no remainder say
if have no remainder say g
g(
(x
x) divides
) divides f
f(
(x
x)
)
 if
if g
g(
(x
x) has no divisors other than itself & 1
) has no divisors other than itself & 1
say it is
say it is irreducible
irreducible (or prime) polynomial
(or prime) polynomial
 arithmetic modulo an irreducible
arithmetic modulo an irreducible
polynomial forms a field
polynomial forms a field

Polynomial GCD
Polynomial GCD
 can find greatest common divisor for polys
can find greatest common divisor for polys

c(x)
c(x) = GCD(
= GCD(a(x), b(x)
a(x), b(x)) if
) if c(x)
c(x) is the poly of greatest
is the poly of greatest
degree which divides both
degree which divides both a(x), b(x)
a(x), b(x)
 can adapt Euclid’s Algorithm to find it:
can adapt Euclid’s Algorithm to find it:
EUCLID[
EUCLID[a
a(
(x
x)
), b
, b(
(x
x)]
)]
1.
1. A(
A(x
x) =
) = a
a(
(x
x); B(
); B(x
x) =
) = b
b(
(x
x)
)
2. if
2. if B(
B(x
x) = 0
) = 0 return
return A(
A(x
x) = gcd[
) = gcd[a
a(
(x
x)
), b
, b(
(x
x)]
)]
3.
3. R(
R(x
x) = A(
) = A(x
x) mod B(
) mod B(x
x)
)
4.
4. A(
A(x
x) ¨ B(
) ¨ B(x
x)
)
5.
5. B(
B(x
x) ¨ R(
) ¨ R(x
x)
)
6. goto
6. goto 2
2

Modular Polynomial
Modular Polynomial
Arithmetic
Arithmetic
 can compute in field GF(2
can compute in field GF(2n
n
)
)

polynomials with coefficients modulo 2
polynomials with coefficients modulo 2

whose degree is less than n
whose degree is less than n

hence must reduce modulo an irreducible poly
hence must reduce modulo an irreducible poly
of degree n (for multiplication only)
of degree n (for multiplication only)
 form a finite field
form a finite field
 can always find an inverse
can always find an inverse

can extend Euclid’s Inverse algorithm to find
can extend Euclid’s Inverse algorithm to find

Example GF(2
Example GF(23
3
)
)

Computational
Computational
Considerations
Considerations
 since coefficients are 0 or 1, can represent
since coefficients are 0 or 1, can represent
any such polynomial as a bit string
any such polynomial as a bit string
 addition becomes XOR of these bit strings
addition becomes XOR of these bit strings
 multiplication is shift & XOR
multiplication is shift & XOR

cf long-hand multiplication
cf long-hand multiplication
 modulo reduction done by repeatedly
modulo reduction done by repeatedly
substituting highest power with remainder
substituting highest power with remainder
of irreducible poly (also shift & XOR)
of irreducible poly (also shift & XOR)

Computational Example
Computational Example
 in
in GF(2
GF(23
3
) have
) have (x
(x2
2
+1) is 101
+1) is 1012
2 & (x
& (x2
2
+x+1) is 111
+x+1) is 1112
2
 so addition is
so addition is

(x
(x2
2
+1) + (x
+1) + (x2
2
+x+1) = x
+x+1) = x

101 XOR 111 = 010
101 XOR 111 = 0102
2
 and multiplication is
and multiplication is

(x+1).(x
(x+1).(x2
2
+1) = x.(x
+1) = x.(x2
2
+1) + 1.(x
+1) + 1.(x2
2
+1)
+1)
= x
= x3
3
+x+x
+x+x2
2
+1 = x
+1 = x3
3
+x
+x2
2
+x+1
+x+1

011.101 = (101)<<1 XOR (101)<<0 =
011.101 = (101)<<1 XOR (101)<<0 =
1010 XOR 101 = 1111
1010 XOR 101 = 11112
2
 polynomial modulo reduction (get q(x) & r(x)) is
polynomial modulo reduction (get q(x) & r(x)) is

(x
(x3
3
+x
+x2
2
+x+1 ) mod (x
+x+1 ) mod (x3
3
+x+1) = 1.(x
+x+1) = 1.(x3
3
+x+1) + (x
+x+1) + (x2
2
) = x
) = x2
2

1111 mod 1011 = 1111 XOR 1011 = 0100
1111 mod 1011 = 1111 XOR 1011 = 01002
2

Using a Generator
Using a Generator
 equivalent definition of a finite field
equivalent definition of a finite field
 a
a generator
generator g is an element whose
g is an element whose
powers generate all non-zero elements
powers generate all non-zero elements

in F have 0, g
in F have 0, g0
0
, g
, g1
1
, …, g
, …, gq-2
q-2
 can create generator from
can create generator from root
root of the
of the
irreducible polynomial
irreducible polynomial
 then implement multiplication by adding
then implement multiplication by adding
exponents of generator
exponents of generator

Summary
Summary
 have considered:
have considered:

concept of groups, rings, fields
concept of groups, rings, fields

modular arithmetic with integers
modular arithmetic with integers

Euclid’s algorithm for GCD
Euclid’s algorithm for GCD

finite fields GF(p)
finite fields GF(p)

polynomial arithmetic in general and in GF(2
polynomial arithmetic in general and in GF(2n
n
)
)

ch04 cryptography1cryptography1cryptography1

More Related Content

Similar to ch04 cryptography1cryptography1cryptography1

Recently uploaded

ch04 cryptography1cryptography1cryptography1

Editor's Notes