Basic Concepts in Number Theory and Finite Fields for Cryptography
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4
Fifth Edition
by William Stallings
Lecture slides by Lawrie Brown
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The next morning at daybreak, Star flew indoors, seemingly keen for
a lesson. I said, "Tap eight." She did 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 for her nut. It is astonishing that Star learned to
count up to 8 with no difficulty, and of her own accord discovered
that each number could be given with various different divisions, this
leaving no doubt that she was consciously thinking each number. In
fact, she did mental arithmetic, although unable, like humans, to
name the numbers. But she learned to recognize their spoken
names almost immediately and was able to remember the sounds of
the names. Star is unique as a wild bird, who of her own free will
pursued the science of numbers with keen interest and astonishing
intelligence.
—
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will now introduce finite fields
of increasing importance in cryptography
AES, Elliptic Curve, IDEA, Public Key
concern operations on “numbers”
where what constitutes a “number” and the
type of operations varies considerably
start with basic number theory concepts
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a
=
m
b
(
a
,
b
,
m
a
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g
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s
)
that is
b
divides into
a
with no remainder
denote this
b|a
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a
eg. all of 1,2,3,4,6,8,12,24 divide 24
eg.
13 | 182; –5 | 30; 17 | 289; –3 | 33; 17 | 0
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If
a|1, then a = ±1.
If a|b and b|a, then a = ±b.
Any b /= 0 divides 0.
If a | b and b | c, then a | c
e.g.
11 | 66 and 66 | 198 x 11 | 198
If
b|g and b|h, then b|(mg + nh)
for arbitrary integers m and n
e.g. b = 7; g = 14; h = 63; m = 3; n = 2
hence 7|14 and 7|63
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A
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if divide a by n get integer quotient
q
and
integer remainder
r
such that:
a = qn + r
where
0 <= r < n; q = floor(a/n)
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(
G
C
D
)
a common problem in number theory
GCD (a,b) of a and b is the largest integer
that divides evenly into both a and b
eg GCD(60,24) = 12
define gcd(0, 0) = 0
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e
eg GCD(8,15) = 1
hence 8 & 15 are relatively prime
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C
D
(
1
9
7
0
,
1
0
6
6
)
1970 = 1 x 1066 + 904
gcd(1066, 904)
1066 = 1 x 904 + 162
gcd(904, 162)
904 = 5 x 162 + 94
gcd(162, 94)
162 = 1 x 94 + 68
gcd(94, 68)
94 = 1 x 68 + 26
gcd(68, 26)
68 = 2 x 26 + 16
gcd(26, 16)
26 = 1 x 16 + 10
gcd(16, 10)
16 = 1 x 10 + 6
gcd(10, 6)
10 = 1 x 6 + 4
gcd(6, 4)
6 = 1 x 4 + 2
gcd(4, 2)
4 = 2 x 2 + 0
gcd(2, 0)
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(
1
1
6
0
7
1
8
1
7
4
,
3
1
6
2
5
8
2
5
0
)
Dividend
Dividend
Divisor
Divisor
Quotient
Quotient
Remainder
Remainder
a = 1160718174
a = 1160718174
b = 316258250
b = 316258250
q1 = 3
q1 = 3
r1 = 211943424
r1 = 211943424
b = 316258250
b = 316258250
r1 = 211943424
r1 = 211943424
q2 = 1
q2 = 1
r2 = 104314826
r2 = 104314826
r1 = 211943424
r1 = 211943424
r2 = 104314826
r2 = 104314826
q3 = 2
q3 = 2
r3 = 3313772
r3 = 3313772
r2 = 104314826
r2 = 104314826
r3 = 3313772
r3 = 3313772
q4 = 31
q4 = 31
r4 = 1587894
r4 = 1587894
r3 = 3313772
r3 = 3313772
r4 = 1587894
r4 = 1587894
q5 = 2
q5 = 2
r5 = 137984
r5 = 137984
r4 = 1587894
r4 = 1587894
r5 = 137984
r5 = 137984
q6 = 11
q6 = 11
r6 = 70070
r6 = 70070
r5 = 137984
r5 = 137984
r6 = 70070
r6 = 70070
q7 = 1
q7 = 1
r7 = 67914
r7 = 67914
r6 = 70070
r6 = 70070
r7 = 67914
r7 = 67914
q8 = 1
q8 = 1
r8 = 2516
r8 = 2516
r7 = 67914
r7 = 67914
r8 = 2516
r8 = 2516
q9 = 31
q9 = 31
r9 = 1078
r9 = 1078
r8 = 2516
r8 = 2516
r9 = 1078
r9 = 1078
q10 = 2
q10 = 2
r10 = 0
r10 = 0
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since with integers can always write:
a = qn + b
usually chose smallest positive remainder as residue
•
ie.
0 <= b <= n-1
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•
eg. -12 mod 7
=
-5 mod 7
=
2 mod 7
=
9 mod 7
a
&
b
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n
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r
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t
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f
:
a
m
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d
n
=
b
m
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d
n
when divided by
n,
a & b have same remainder
eg. 100 = 34 mod 11
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can perform arithmetic with residues
uses a finite number of values, and loops
back from either end
Z
n
= {0, 1, . . . , (n – 1)}
modular arithmetic is when do addition &
multiplication and modulo reduce answer
can do reduction at any point, ie
a+b mod n = [a mod n + b mod n] mod n
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1.
[(a mod n) + (b mod n)] mod n
= (a + b) mod n
2.
[(a mod n) – (b mod n)] mod n
= (a – b) mod n
3.
[(a mod n) x (b mod n)] mod n
= (a x b) mod n
e.g.
[(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2 (11 + 15) mod 8 = 26 mod 8 = 2
[(11 mod 8) – (15 mod 8)] mod 8 = –4 mod 8 = 4 (11 – 15) mod 8 = –4 mod 8 = 4
[(11 mod 8) x (15 mod 8)] mod 8 = 21 mod 8 = 5 (11 x 15) mod 8 = 165 mod 8 = 5
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an efficient way to find the GCD(a,b)
uses theorem that:
GCD(a,b) = GCD(b, a mod b)
Euclidean Algorithm to compute GCD(a,b) is:
Euclid(a,b)
if (b=0) then return a;
else return Euclid(b, a mod b);
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calculates not only GCD but x & y:
ax + by = d = gcd(a, b)
useful for later crypto computations
follow sequence of divisions for GCD but
assume at each step i, can find x &y:
r = ax + by
at end find GCD value and also x & y
if GCD(a,b)=1 these values are inverses
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EXTENDED EUCLID(
m
,
b
)
1.
(A1, A2, A3)=(1, 0,
m
);
(B1, B2, B3)=(0, 1,
b
)
2. if
B3 = 0
return
A3 = gcd(
m
,
b
); no inverse
3. if
B3 = 1
return
B3 = gcd(
m
,
b
); B2 =
b
–1
mod
m
4.
Q = A3 div B3
5.
(T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3)
6.
(A1, A2, A3)=(B1, B2, B3)
7.
(B1, B2, B3)=(T1, T2, T3)
8. goto
2
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f
5
5
0
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G
F
(
1
7
5
9
)
G
r
o
u
p
a set of elements or “numbers”
may be finite or infinite
with some operation whose result is also
in the set (closure)
obeys:
associative law:
(a.b).c = a.(b.c)
has identity
e
:
e.a = a.e = a
has inverses
a
-1
:
a.a
-1
= e
if commutative
a.b = b.a
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example:
a
-3
= a.a.a
and let identity be:
e=
a
0
a group is cyclic if every element is a
power of some fixed element
ie
b =
a
k
for some
a
and every
b
in group
a
is said to be a generator of the group
R
i
n
g
a set of “numbers”
with two operations (addition and multiplication)
which form:
an abelian group with addition operation
and multiplication:
has closure
is associative
distributive over addition:
a(b+c) = ab + ac
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a
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F
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l
d
a set of numbers
with two operations which form:
abelian group for addition
abelian group for multiplication (ignoring 0)
ring
have hierarchy with more axioms/laws
group -> ring -> field
G
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,
R
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g
,
F
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d
F
i
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e
(
G
a
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o
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s
)
F
i
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l
d
s
finite fields play a key role in cryptography
c
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t
b
e
a
p
o
w
e
r
o
f
a
p
r
i
m
e
p
n
known as Galois fields
denoted GF(p
n
)
in particular often use the fields:
GF(p)
GF(2
n
)
G
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o
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s
F
i
e
l
d
s
G
F
(
p
)
GF(p) is the set of integers {0,1, … , p-1}with arithmetic operations modulo prime p
these form a finite field
since have multiplicative inverses
find inverse with Extended Euclidean algorithm
hence arithmetic is “well-behaved” and can
do addition, subtraction, multiplication, and
division without leaving the field GF(p)
G
F
(
7
)
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A
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c
can compute using polynomials
f
(
x
) = a
n
x
n
+ a
n-1
x
n-1
+ … + a
1
x +
a
0
= ∑ a
i
x
i
•
nb. not interested in any specific value of x
•
which is known as the indeterminate
several alternatives available
ordinary polynomial arithmetic
poly arithmetic with coords mod p
poly arithmetic with coords mod p and
polynomials mod m(x)
O
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A
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h
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t
i
c
add or subtract corresponding coefficients
multiply all terms by each other
eg
let
f
(
x
) =
x
3
+
x
2
+ 2 and
g
(
x
) =
x
2
–
x
+ 1
f
(
x
) +
g
(
x
) =
x
3
+ 2
x
2
–
x
+ 3
f
(
x
) –
g
(
x
) =
x
3
+
x
+ 1
f
(
x
) x
g
(
x
) =
x
5
+ 3
x
2
– 2
x
+ 2
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C
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f
f
i
c
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n
t
s
when computing value of each coefficient
do calculation modulo some value
forms a polynomial ring
could be modulo any prime
but we are most interested in mod 2
ie all coefficients are 0 or 1
eg. let
f
(
x
) =
x
3
+
x
2
and
g
(
x
) =
x
2
+
x
+ 1
f
(
x
) +
g
(
x
) =
x
3
+
x
+ 1
f
(
x
) x
g
(
x
) =
x
5
+
x
2
P
o
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y
n
o
m
i
a
l
D
i
v
i
s
i
o
n
can write any polynomial in the form:
f
(
x
) =
q
(
x
)
g
(
x
) +
r
(
x
)
can interpret
r
(
x
)
as being a remainder
r
(
x
) =
f
(
x
) mod
g
(
x
)
if have no remainder say
g
(
x
) divides
f
(
x
)
i
f
g
(
x
)
h
a
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s
o
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s
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t
h
e
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t
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a
n
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s
e
l
f
&
1
s
a
y
i
t
i
s
i
r
r
e
d
u
c
i
b
l
e
(
o
r
p
r
i
m
e
)
p
o
l
y
n
o
m
i
a
l
arithmetic modulo an irreducible
polynomial forms a field
P
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y
n
o
m
i
a
l
G
C
D
can find greatest common divisor for polys
c(x)
= GCD(
a(x), b(x)
) if
c(x)
is the poly of greatest
degree which divides both
a(x), b(x)
can adapt Euclid’s Algorithm to find it:
Euclid(
a
(
x
)
, b
(
x
)
)
if (
b
(
x
)
=0) then return
a
(
x
)
;
else return
Euclid(
b
(
x
)
,
a
(
x
)
mod
b
(
x
)
);
all foundation for polynomial fields as see next
M
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P
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m
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A
r
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h
m
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t
i
c
can compute in field GF(2
n
)
polynomials with coefficients modulo 2
whose degree is less than n
hence must reduce modulo an irreducible poly
of degree n (for multiplication only)
form a finite field
can always find an inverse
can extend Euclid’s Inverse algorithm to find
E
x
a
m
p
l
e
G
F
(
2
3
)
C
o
m
p
u
t
a
t
i
o
n
a
l
C
o
n
s
i
d
e
r
a
t
i
o
n
s
since coefficients are 0 or 1, can represent
any such polynomial as a bit string
addition becomes XOR of these bit strings
multiplication is shift & XOR
cf long-hand multiplication
modulo reduction done by repeatedly
substituting highest power with remainder
of irreducible poly (also shift & XOR)
C
o
m
p
u
t
a
t
i
o
n
a
l
E
x
a
m
p
l
e
in
GF(2
3
) have
(x
2
+1) is 101
2
& (x
2
+x+1) is 111
2
so addition is
(x
2
+1) + (x
2
+x+1) = x
101 XOR 111 = 010
2
and multiplication is
(x+1).(x
2
+1) = x.(x
2
+1) + 1.(x
2
+1)
= x
3
+x+x
2
+1 = x
3
+x
2
+x+1
011.101 = (101)<<1 XOR (101)<<0 =
1010 XOR 101 = 1111
2
polynomial modulo reduction (get q(x) & r(x)) is
(x
3
+x
2
+x+1 ) mod (x
3
+x+1) = 1.(x
3
+x+1) + (x
2
) = x
2
1111 mod 1011 = 1111 XOR 1011 = 0100
2
U
s
i
n
g
a
G
e
n
e
r
a
t
o
r
equivalent definition of a finite field
a
g
e
n
e
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a
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g
i
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a
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a
l
l
n
o
n
-
z
e
r
o
e
l
e
m
e
n
t
s
in F have 0, g
0
, g
1
, …, g
q-2
c
a
n
c
r
e
a
t
e
g
e
n
e
r
a
t
o
r
f
r
o
m
r
o
o
t
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f
t
h
e
i
r
r
e
d
u
c
i
b
l
e
p
o
l
y
n
o
m
i
a
l
then implement multiplication by adding
exponents of generator
S
u
m
m
a
r
y
have considered:
divisibility & GCD
modular arithmetic with integers
concept of groups, rings, fields
Euclid’s algorithm for GCD & Inverse
finite fields GF(p)
polynomial arithmetic in general and in GF(2
n
)
Lecture slides by Lawrie Brown for “Cryptography and Network Security”, 5/e, by William Stallings, Chapter Chapter 4 – “Basic Concepts in Number Theory and Finite Fields”.
In this presentation, the basic concepts of number theory, including divisors, properties of divisibility, the division algorithm, and the greatest common divisor (GCD), are explored in the context of cryptography and network security. The importance of finite fields in cryptographic operations such as AES, Elliptic Curve, IDEA, and Public Key encryption is highlighted through practical examples and insights. Through engaging narratives and mathematical explanations, readers gain a deeper understanding of foundational mathematical principles essential for secure communication in the digital realm.
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Cryptography and Network Security Chapter 4 Fifth Edition by William Stallings Lecture slides by Lawrie Brown
Chapter 4 Basic Concepts in Number Theory and Finite Fields The next morning at daybreak, Star flew indoors, seemingly keen for a lesson. I said, "Tap eight." She did 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 for her nut. It is astonishing that Star learned to count up to 8 with no difficulty, and of her own accord discovered that each number could be given with various different divisions, this leaving no doubt that she was consciously thinking each number. In fact, she did mental arithmetic, although unable, like humans, to name the numbers. But she learned to recognize their spoken names almost immediately and was able to remember the sounds of the names. Star is unique as a wild bird, who of her own free will pursued the science of numbers with keen interest and astonishing intelligence. Living with Birds, Len Howard
Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on numbers where what constitutes a number and the type of operations varies considerably start with basic number theory concepts
Divisors say a non-zero number b divides a if for some m have a=mb (a,b,m all integers) that is b divides into a with no remainder denote this b|a and say that b is a divisor of a eg. all of 1,2,3,4,6,8,12,24 divide 24 eg. 13 | 182; 5 | 30; 17 | 289; 3 | 33; 17 | 0
Properties of Divisibility If a|1, then a = 1. If a|b and b|a, then a = b. Any b /= 0 divides 0. If a | b and b | c, then a | c e.g. 11 | 66 and 66 | 198 x 11 | 198 If b|g and b|h, then b|(mg + nh) for arbitrary integers m and n e.g. b = 7; g = 14; h = 63; m = 3; n = 2 hence 7|14 and 7|63
Division Algorithm if divide a by n get integer quotient q and integer remainder r such that: a = qn + r where 0 <= r < n; q = floor(a/n) remainder r often referred to as a residue
Greatest Common Divisor (GCD) a common problem in number theory GCD (a,b) of a and b is the largest integer that divides evenly into both a and b eg GCD(60,24) = 12 define gcd(0, 0) = 0 often want no common factors (except 1) define such numbers as relatively prime eg GCD(8,15) = 1 hence 8 & 15 are relatively prime
Example GCD(1970,1066) 1970 = 1 x 1066 + 904 1066 = 1 x 904 + 162 904 = 5 x 162 + 94 gcd(162, 94) 162 = 1 x 94 + 68 94 = 1 x 68 + 26 68 = 2 x 26 + 16 26 = 1 x 16 + 10 16 = 1 x 10 + 6 10 = 1 x 6 + 4 6 = 1 x 4 + 2 4 = 2 x 2 + 0 gcd(1066, 904) gcd(904, 162) gcd(94, 68) gcd(68, 26) gcd(26, 16) gcd(16, 10) gcd(10, 6) gcd(6, 4) gcd(4, 2) gcd(2, 0)
GCD(1160718174, 316258250) Dividend a = 1160718174 b = 316258250 q1 = 3 b = 316258250 r1 = 211943424 q2 = 1 r1 = 211943424 r2 = 104314826 q3 = 2 r2 = 104314826 r3 = 3313772 r3 = 3313772 r4 = 1587894 r4 = 1587894 r5 = 137984 r5 = 137984 r6 = 70070 r6 = 70070 r7 = 67914 r7 = 67914 r8 = 2516 r8 = 2516 r9 = 1078 Divisor Quotient Remainder r1 = 211943424 r2 = 104314826 r3 = 3313772 q4 = 31 r4 = 1587894 q5 = 2 r5 = 137984 q6 = 11 r6 = 70070 q7 = 1 r7 = 67914 q8 = 1 r8 = 2516 q9 = 31 r9 = 1078 q10 = 2 r10 = 0
Modular Arithmetic define modulo operator a mod n to be remainder when a is divided by n where integer n is called the modulus b is called a residue of a mod n since with integers can always write: a = qn + b usually chose smallest positive remainder as residue ie. 0 <= b <= n-1 process is known as modulo reduction eg. -12 mod 7 = -5 mod 7 = 2 mod 7 = 9 mod 7 a & b are congruent if: a mod n = b mod n when divided by n, a & b have same remainder eg. 100 = 34 mod 11
Modular Arithmetic Operations can perform arithmetic with residues uses a finite number of values, and loops back from either end Zn = {0, 1, . . . , (n 1)} modular arithmetic is when do addition & multiplication and modulo reduce answer can do reduction at any point, ie a+b mod n = [a mod n + b mod n] mod n
Modular Arithmetic Operations 1. [(a mod n) + (b mod n)] mod n = (a + b) mod n 2. [(a mod n) (b mod n)] mod n = (a b) mod n 3. [(a mod n) x (b mod n)] mod n = (a x b) mod n e.g. [(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2 (11 + 15) mod 8 = 26 mod 8 = 2 [(11 mod 8) (15 mod 8)] mod 8 = 4 mod 8 = 4 (11 15) mod 8 = 4 mod 8 = 4 [(11 mod 8) x (15 mod 8)] mod 8 = 21 mod 8 = 5 (11 x 15) mod 8 = 165 mod 8 = 5
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
Modulo 8 Multiplication + 0 1 2 3 4 5 6 7 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 2 0 2 4 6 0 2 4 6 3 0 3 6 1 4 7 2 5 4 0 4 0 4 0 4 0 4 5 0 5 2 7 4 1 6 3 6 0 6 4 2 0 6 4 2 7 0 7 6 5 4 3 2 1
Euclidean Algorithm an efficient way to find the GCD(a,b) uses theorem that: GCD(a,b) = GCD(b, a mod b) Euclidean Algorithm to compute GCD(a,b) is: Euclid(a,b) if (b=0) then return a; else return Euclid(b, a mod b);
Extended Euclidean Algorithm calculates not only GCD but x & y: ax + by = d = gcd(a, b) useful for later crypto computations follow sequence of divisions for GCD but assume at each step i, can find x &y: r = ax + by at end find GCD value and also x & y if GCD(a,b)=1 these values are inverses
Finding Inverses EXTENDED EUCLID(m, b) 1. (A1, A2, A3)=(1, 0, m); (B1, B2, B3)=(0, 1, b) 2. if B3 = 0 return A3 = gcd(m, b); no inverse 3. if B3 = 1 return B3 = gcd(m, b); B2 = b 1 mod m 4. Q = A3 div B3 5. (T1, T2, T3)=(A1 Q B1, A2 Q B2, A3 Q B3) 6. (A1, A2, A3)=(B1, B2, B3) 7. (B1, B2, B3)=(T1, T2, T3) 8. goto 2
Inverse of 550 in GF(1759) Q A1 1 A2 0 A3 1759 B1 0 B2 1 B3 550 3 5 21 0 1 1 550 109 5 1 5 106 3 16 109 5 4 3 16 5 339 1 106 339 4 111 355 1
Group a set of elements or numbers may be finite or infinite with some operation whose result is also in the set (closure) obeys: associative law: (a.b).c = a.(b.c) has identity e: e.a = a.e = a has inverses a-1: a.a-1 = e if commutative a.b = b.a then forms an abelian group
Cyclic Group define exponentiation as repeated application of operator example: a-3 = a.a.a and let identity be: e=a0 a group is cyclic if every element is a power of some fixed element ie b =ak for some a and every b in group a is said to be a generator of the group
Ring a set of numbers with two operations (addition and multiplication) which form: an abelian group with addition operation and multiplication: has closure is associative distributive over addition: if multiplication operation is commutative, it forms a commutative ring if multiplication operation has an identity and no zero divisors, it forms an integral domain a(b+c) = ab + ac
Field a set of numbers with two operations which form: abelian group for addition abelian group for multiplication (ignoring 0) ring have hierarchy with more axioms/laws group -> ring -> field
Finite (Galois) Fields finite fields play a key role in cryptography can show number of elements in a finite field must be a power of a prime pn known as Galois fields denoted GF(pn) in particular often use the fields: GF(p) GF(2n)
Galois Fields GF(p) GF(p) is the set of integers {0,1, , p-1} with arithmetic operations modulo prime p these form a finite field since have multiplicative inverses find inverse with Extended Euclidean algorithm hence arithmetic is well-behaved and can do addition, subtraction, multiplication, and division without leaving the field GF(p)
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
Polynomial Arithmetic can compute using polynomials f(x) = anxn + an-1xn-1+ + a1x + a0= aixi nb. not interested in any specific value of x which is known as the indeterminate several alternatives available ordinary polynomial arithmetic poly arithmetic with coords mod p poly arithmetic with coords mod p and polynomials mod m(x)
Ordinary Polynomial Arithmetic add or subtract corresponding coefficients multiply all terms by each other eg let f(x) = x3 + x2 + 2 and g(x) = x2 x + 1 f(x) + g(x) = x3 + 2x2 x + 3 f(x) g(x) = x3 + x + 1 f(x) x g(x) = x5 + 3x2 2x + 2
Polynomial Arithmetic with Modulo Coefficients when computing value of each coefficient do calculation modulo some value forms a polynomial ring could be modulo any prime but we are most interested in mod 2 ie all coefficients are 0 or 1 eg. let f(x) = x3 + x2 and g(x) = x2 + x + 1 f(x) + g(x) = x3 + x + 1 f(x) x g(x) = x5 + x2
Polynomial Division can write any polynomial in the form: f(x) = q(x) g(x) + r(x) can interpret r(x) as being a remainder r(x) = f(x) mod g(x) if have no remainder say g(x) divides f(x) if g(x) has no divisors other than itself & 1 say it is irreducible (or prime) polynomial arithmetic modulo an irreducible polynomial forms a field
Polynomial GCD can find greatest common divisor for polys c(x) = GCD(a(x), b(x)) if c(x) is the poly of greatest degree which divides both a(x), b(x) can adapt Euclid s Algorithm to find it: Euclid(a(x), b(x)) if (b(x)=0) then return a(x); else return Euclid(b(x), a(x) mod b(x)); all foundation for polynomial fields as see next
Modular Polynomial Arithmetic can compute in field GF(2n) polynomials with coefficients modulo 2 whose degree is less than n hence must reduce modulo an irreducible poly of degree n (for multiplication only) form a finite field can always find an inverse can extend Euclid s Inverse algorithm to find
Computational Considerations since coefficients are 0 or 1, can represent any such polynomial as a bit string addition becomes XOR of these bit strings multiplication is shift & XOR cf long-hand multiplication modulo reduction done by repeatedly substituting highest power with remainder of irreducible poly (also shift & XOR)
Computational Example in GF(23) have (x2+1) is 1012 & (x2+x+1) is 1112 so addition is (x2+1) + (x2+x+1) = x 101 XOR 111 = 0102 and multiplication is (x+1).(x2+1) = x.(x2+1) + 1.(x2+1) = x3+x+x2+1 = x3+x2+x+1 011.101 = (101)<<1 XOR (101)<<0 = 1010 XOR 101 = 11112 polynomial modulo reduction (get q(x) & r(x)) is (x3+x2+x+1 ) mod (x3+x+1) = 1.(x3+x+1) + (x2) = x2 1111 mod 1011 = 1111 XOR 1011 = 01002
Using a Generator equivalent definition of a finite field a generator g is an element whose powers generate all non-zero elements in F have 0, g0, g1, , gq-2 can create generator from root of the irreducible polynomial then implement multiplication by adding exponents of generator
Summary have considered: divisibility & GCD modular arithmetic with integers concept of groups, rings, fields Euclid s algorithm for GCD & Inverse finite fields GF(p) polynomial arithmetic in general and in GF(2n)
