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Title: There%20are%20those%20who%20are%20destined%20to%20be%20good,%20but%20never%20to%20experience%20it.%20I%20believe%20I%20am%20one%20of%20them.


1
  • "There are those who are destined to be good,
    but never to experience it. I believe I am one of
    them."
  • --- Evariste Galois (1811-1832)

2
Mathematical Background A Revision
  • finite fields (FF)
  • required for understanding
  • AES
  • Elliptic Curve Cryptography
  • To study FF, we shall revise the concepts of
  • groups, rings, fields from abstract algebra
  • Modular arithmetic and Euclidean Algorithm
  • Finite fields of the form GF(p), where p is a
    prime number

3
Group Theory History
  • Groups First used by Evariste Galois (b.1811-
    d.1832) in his work, without defining a Group
  • Galois, a student of M. Vernier in 1827 and
  • a contemporary of Cauchy, Poisson, Abel,
    Jacobi, Fourier, Gauss and Napolean (ruled during
    1800-1815)
  • He failed to join Ecole Polytechnique, though he
    appeared twice in the entrance tests.
  • An ardent Republican, he was sent to prison twice
    by the King.

4
Quest for Academy Award
  • 1829 Galois (only 18 years old) submitted two
    papers to Académie des Sciences for publication
    in its Memoirs Cauchy was the referee for the
    papers.
  • Galois read a posthumous paper of Abel and found
    that there was an overlap between his and Abels
    work. So he consulted Cauchy. Cauchy (winner of
    Grand prix in 1816) advised him to rewrite it and
    submit it for Grand Prix.
  • Feb 1830 Galois submitted the modified paper to
    Fourier for Grand Prix Fourier died in April
    1830 and the paper was lost Abel and Jacobi got
    the Grand Prix prize.

5
Last Night
  • 1831 Galois again submitted to Académie des
    Sciences Poisson was the Reviewer. He did not
    understand the paper and rejected it.
  • night of 30 May 1832 injured at the duel with
    Perscheux d'Herbinville over the prisons
    physicians daughter named Stephanie-Felice du
    Motel abandoned by both Perscheux as well as his
    seconds. A peasant took him to a hospital, where
    he died at the age of 21 in 1832.
  • A story? an injured Galois wrote notes on the
    rejected paper a night of furious writings by
    Galois

6
First definitions
  • Liouville, Galoiss elder brother, copied his
    papers and sent them to Gauss, Jacobi and others
  • 14 years later
  • 1846 Liouville got Galois' papers published
  • 1845 Cauchy defined a "conjugate system of
    substitutions, another name of Groups. During
    1845-46, he wrote 25 papers on it.
  • 1854 The first person to try to give (not
    completely correct) an abstract definition of a
    group Cayley.
  • 1863 Jordans commentary on Galois paper and his
    book used the term GROUP

7
Group Theory the first
modern book
  • Walter Ledermann's book Introduction to the
    theory of finite groups, published by publisher
    Oliver Boyd in Edinburgh
  • 1949 (when Ledermann was 38 years old,
    assistant lecturer at St Andrews )
  • was based on Schur's lectures on group theory.

8
Group Theory and communism
  • Ledermann wrote it in the British Museum Library
    (sitting in the same chair where Karl Marx wrote
    Das Capital)
  • Ledermann came for a lecture on Group Theory at
    University of Notre Dame in the United States
    the parcel of books was stopped by US Customs,
    who mistook it as a book of Communist groups,
    till the Head of Dept of Notre Dame personally
    spoke to Customs.

9
A note on types of numbers
  • Positive integers and Integers
  • Rational numbers A rational number is any
    number that can be written as a ratio of two
    integers. Reference 1 http//bing.search.symp
    atico.ca/?qdifference20between20a20real20numb
    er20and20a20rational20numbermkten-casetLang
    en-CA
  •  Examples Integers, fractions, mixed numbers,
    and decimals together with their negative
    images.
  • Examples of irrational numbers v2, v3, v5, pi
    (p), e
  • p a mathematical constant whose value is the
    ratio of any circle 's circumference to its
    diameter 3.14159265358979323846264338327950288419
    716939937510
  • e base of the natural logarithm known as
    Napier's constant symbol honors Euler
  • 2.718281828459045235360287471352662497757
    .
  • is the unique number with the property that
    the area of the region bounded by the hyperbola
    y 1/x, the x-axis, and the vertical lines x 1
    and x e is 1. In other words

  • 1?e (dx/x) ln e 1.

10
A note on types of numbers..2
  • Real numbers
  • Any number that can be found on the number line
  • a number required to label any point on the
    number line
  • a number whose absolute value names the distance
    of any point from 0.
  • both rational and irrational numbers
  • Between any two rational numbers on the number
    line there is an irrational number. 1
  • Between any two irrational numbers there is a
    rational number 1

11
A note on types of numbers..3
  • Complex numbers Example x i y ,
  • where
  • x and y real numbers and
  • i v(-1) .
  • The field of complex numbers includes the field
    of real numbers as a subfield.
  • References (i) http//www.themathpage.com/aPreCal
    c/rational-irrational-numbers.htm
  • (ii) http//mathworld.wolfram.com/ComplexNumber.ht
    ml

12
Group
  • DEFINITION
  • a set of elements or numbers
  • with some operation whose result is also in the
    set (closure)
  • (The operation is shown through the symbol .
    in the examples below.)
  • obeys
  • associative law (a.b).c a.(b.c)
  • has an identity element e so that for all
  • a ? G, e.a a.e a
  • For each a ? G, there exists an inverse element
    a-1 ? G,such that a.a-1 e

13
Example of a group
  • Example 1 N a set of n distinct symbols
  • 1,2,..,n
  • S set of all permutations of the n symbols
  • S is a Group, under the operation of permutation.
  • Prove
  • Closure
  • Association
  • Existence of an identity element as a member of
    the group
  • Existence of an inverse for every member of the
    Group
  • A Finite Group if the number of members of the
    group
  • is finite.
  • An Infinite Group

14
Abelian Group
  • If in addition to the three properties stated in
  • slide 2, the property of commutation is
  • satisfied, G is said to be an abelian group.
  • Commutative if for all a,b ? G,
  • a.b b.a
  • Examples 2. Prove that S, as defined in
  • Example 1, is not an Abelian group.
  • 3. Prove that the set of integers (positive,
  • negative and zero) is an Abelian group under
  • addition. Hint Identity element 0, Inverse
  • element of X is X.

15
Some Definitions and the definition of a
Cyclic Group
  • Exponentiation defined as repeated application
    of an operator.
  • example a3 a.a.a
  • Identity Element ea0
  • If a be the inverse of a, a-n (a)n
  • A Group is cyclic if every member of the Group is
    generated by a single element a, (called the
    Generator) through exponentiation. a is a
    member of the Group.
  • A cyclic group is Abelian.

16
Cyclic Group (continued)
  • Cyclic group
  • b ak
  • For some integer value of k, b should stand
    for every member of the Group
  • A cyclic Group may be finite or infinite.
  • Subgroups of a cyclic group are also cyclic.
  • A cyclic group may have more than one generator
    element.
  • Example 4a A group of integers, under the
    operation of addition, is a cyclic group. Both 1
    and 1 are the generators.

17
Cyclic Groups of Finite Group Order
  • A cyclic group of finite group order n is denoted
    as Cn with a generator element a and an identity
    element e such that e an.
  • The operations of such a group may be defined
    mod n.
  • Example 4b Zn is a finite cyclic group of
    integers 0,1,2(n-1), under the operation of
    addition mod n, with a generator element of 1
    and an identity element of 0

18
Generator of a Field
  • GENERATOR an element whose successive powers
    take on every element of the field except the
    zero
  • For Prime number fields a gj modp
  • Not every element of a field is a generator.
  • For every 0ltjlt(p-1), a different element is
    obtained.
  • ORDER of a generator element the smallest
    exponent j (lt p), that gets the identity element.
  • gj mod p 1

19
Example of a generator and order
  • Examples1 Modulo 13
  • 4 and 5 are NOT generator elements.
  • a 2 is a generator element.
  • Its order is 12.

1 2 3 4 5 6 7 8 9 10 11 12
2 4 8 3 6 12 11 9 5 10 7 1
exponent, b ab mod13
20
Another Example a generator and order
  • Examples 2 Modulo 11 2, 6, 7 and 8 are examples
    of generator elements.
  • Order of 2, 6, 7 and 8 10.

21
Ring
  • Consider a set of numbers with two binary
    operations, called
  • addition and multiplication.
  • If the set constitutes an Abelian group with
    addition operation, and,
  • if with multiplication operation, the set
  • has closure For a, b ? G, a.b ? G
  • is associative For a, b, c ? G, (a.b).c
    a.(b.c)
  • distributive over addition
  • a.(bc) a.b a.c
  • the set constitutes a Ring.
  • In a Ring, we can do multiplication,
  • addition and subtraction without leaving the
    Ring.

22
Commutative Ring
  • Ex 5 The set of all square matrices is a Ring
    over addition and multiplication.
  • For a Ring, if multiplication operation is
    commutative, the set forms a commutative ring.
  • Examples
  • Ex 6 The set of matrices of Ex 5 is NOT a
    commutative Ring.
  • Ex 7 The set S2 of even integers ( positive,
    negative and 0), under the operations of addition
    and multiplication, is a Commutative Ring.

23
Integral Domain
  • A commutative ring R is said to constitute an
    Integral Domain if,
  • multiplication operation has an identity
  • a.1 1.a for all a ? R,
  • and if,
  • for a, b ? R, if a.b 0, then either
  • a 0 or b 0.
  • Ex 8 S3, the set of integers (positive, negative
    and 0) under the operations of addition and
    multiplication is an Integral domain.

24
Field
  • a Field a set of elements F, with two binary
  • operations, called addition and multiplication,
  • such that
  • F is an Integral Domain, and,
  • For each a ? F, except 0, there is an element a-1
    in F such that
  • a. a-1 a-1.a 1
  • (Existence of multiplicative inverse)

25
Field (continued)
  • Thus in a Field, we can do addition, subtraction,
    multiplication and division without leaving the
    set.
  • Ex 9.The set of all integers S3 is not a Field.
  • 10.The following are Fields
  • The set of Rational Numbers
  • The set of real numbers
  • The set of complex numbers.
  • All of the above examples of Fields have infinite
  • number of elements. We shall see that Fields
  • can be finite also.

26
Group, Ring and Field
A1 closure under addition
A2 Associativity of addition
A3 Additive identity
A4 Additive inverse
Group
Abelian Group
A5 Commutativity of addition
M1 closure under multiplication
M2 Associativity of multiplication
Ring
M3 Distributive laws
M4 Commutativity of multiplication
Commutative Ring
M5 Multiplicative identity
Integral domain
M6 No zero divisors
Field
M7 Multiplicative inverse
27
Mathematical properties 1
  • A1 If a and b belong to S, then a b is also in
    S
  • A2 a (bc) (ab) c for all a,b,c in S
  • A3 There is an element 0 in R such that
  • a 0 0 a a for all a in S
  • A4 For each a in S there is an element a in S
  • such that a (-a) (-a) a 0
  • A5 a b b a for all a,b in A
  • M1 If a and b belong to S, then ab is also in S
  • M2 a (bc) (ab) c for all a, b, c in S

28
Mathematical properties 2
  • M3 a(bc) ab ac for all a, b, c in S
  • (ab)c ac bc for all a, b, c in S
  • M4 ab ba for all a, b in S
  • M5 There is an element 1 is S such that
  • a1 1a a for all a in S
  • M6 If a , b in S and ab 0, then either
  • a 0 or b 0
  • M7 If a belongs to S and a ? 0, there is an
  • element a-1 in S such that a. a-1 a-1. a
    1

29
Agenda
  • After defining Rings and Fields
  • Modular arithmetic
  • Divisors, GCD, Euclids theorem
  • prime numbers
  • Fields of type Zp
  • Finite Fields, Extended Euclids Theorem for
  • finding multiplicative inverse
  • Polynomial arithmetic

30
Modular Arithmetic Definitions
  • modulo operator a mod n b
  • where b is the remainder when a is divided by
    n b is called the residue of a mod n.
  • a q.n b 0 lt b lt n q ?a/n?
  • where ?x? is the largest integer
  • less than or equal to x
  • Example 13 a (bc)mod 8
  • In the next slide, b is the element given in the
    first column (outside the box). c is the element
    given in the top row (outside the box).
  • The values of a are given in the box.

31
Modulo 8 Example
32
Congruency mod n
  • If a mod n b mod n, a and b are said to be
    congruent mod n.
  • The above statement may be written as,
  • ab mod n
  • reducing k modulo n The process of finding the
    smallest Non-negative integer, to which k is
    congruent

33
Modular Arithmetic A
Revision (continued)
  • Modular Arithmetic
  • a qn r.

r
0 1.n
2.n q.n a
(q1).n
r
0
-q.n a -(q-1).n
-3.n -2.n -n

Thus 11 1.7 4 ? r 4 11 mod 7
-11 -2.7 3 ? r 3 -11mod 7
34
k mod m
  • 11 mod 7 4
  • (-11) mod 7 3
  • In general, If r k mod m,
  • ( - k) mod m m - r if r ? 0
  • But ( - k) mod m 0 if r 0.
  • i.e. k mod m may or may not be equal to (-k) mod
    m.
  • r k mod m k mod (-m) k mod(lml)

35
Reducing k modulo 7
Example 12
  • ...
  • -21 -20 -19 -18 -17 -16 -15
  • -14 -13 -12 -11 -10 -9 -8
  • -7 -6 -5 -4 -3 -2 -1
  • 0 1 2 3 4 5 6
  • 7 8 9 10 11 12 13
  • 14 15 16 17 18 19 20
  • 21 22 23 24 25 26 27
  • 28 29 30 31 32 33 34
  • ...
  • All the elements in a column are congruent mod 7
  • O .,-21,-14,-7,0,7,14.
  • is called a Residue Class. (Every column
    constitutes a Residue Class.)
  • The Smallest Non-negative integer of the class is
    used to represent the class.

Reduced values
36
Modular Arithmetic
  • a mod n b mod n mod n
  • (a b)mod n
  • a mod n - b mod n mod n
  • (a - b)mod n
  • a mod n x b mod n mod n
  • (a x b)mod n
  • Ex 14 of ExponentiationTo evaluate 1211mod 7
  • 122mod 7 4 128mod 7 44mod 7 4
  • 12 x 122 x 128 mod 7 5 x 4 x 4 mod 7 3

37
  • Note that the positions of primes constitute
    just about the most fundamental, inarguable,
    nontrivial information available to our
    consciousness. This transcends history, culture,
    and opinion. It would appear to exist 'outside'
    space and time and yet to be accessible to any
    consciousness with some sense of repetition,
    rhythm, or counting.
  • --
    Matthew R. Watkins,
  • School of Mathematical Sciences at Exeter
    University, UK http//www.maths.ex.ac.uk/7E
    mwatkins/zeta/ss-b.htm, as of November 3, 2007

38
Modular Arithmetic Additive and multiplicative
inverses
  • additive inverse Let c be the inverse of a.
  • Then a c 0 mod n.
  • Example 15 Additive inverse of 5 mod 8
  • 5 c 0 mod 8. Therefore c 3
  • multiplicative inverse Let c be the
  • inverse of a.
  • Then a x c 1 mod n.
  • Example 16 Multiplicative inverse of 5 mod 8
  • 5 x c 1 mod 8. Therefore c 5, 13, .

39
Relatively Prime Numbers
  • Two integers are said to be relatively prime if
    their only common positive integer factor is 1.
  • In Example 16,
  • 5 and 8 are relatively prime.
  • Consider the case where a and n have a common
    factor other than 1 (i. e. the case where a and
    n are not relatively prime)

40
Multiplicative Inverse (continued)
  • Example 17 a6 n8
  • 6.c 1 mod 8
  • No value of c, that satisfies the above, can be
    found .
  • In general an integer has a multiplicative
    inverse in Zn if that integer is relatively
    prime to n.

41
Inverses for modulo 8
a Additive Inverse of a Multiplicative Inverse of a
0 0 -
1 7 1
2 6 -
3 5 3
4 4 -
5 3 5
6 2 -
7 1 7
42
Multiplicative Inverse Table 2
a 6.a mod 8 5.a mod 8
0 0 0
1 6 5
2 4 2
3 2 7
4 0 4
5 6 1
6 4 6
7 2 3
a 5 is the multiplicative inverse of 5 mod 8.
43
Multiplicative Inverse Table 2
Continued
a 6.a mod 8 5.a mod 8
8 0 0
9 6 5
10 4 2
11 2 7
12 0 4
13 6 1
14 4 6
15 2 3
a 13 is the multiplicative inverse of 5 mod 8.
44
Multiplicative Inverse
  • Let c be the Multiplicative Inverse of b mod n.
  • b.c 1 mod n k.n 1
  • Therefore
  • b.(c n) (k b).n 1
  • k1.n 1
  • Thus c, c n, c 2n. are all multiplicative
    inverses of c. However for a field Zp, with
    members as 0,1,2,3.(p-1), the smallest positive
    number would be said to be the Multiplicative
    Inverse.

45
Some properties of modulo operator
  • some peculiarities
  • if (ab)(ac) mod n then bc mod n
  • but if (a.b)(a.c) mod n then bc mod n
  • only if a is relatively
    prime to n
  • Proof
  • Given (ab) (ac) mod n
  • Add -a (the additive inverse of a) to both sides.
  • -a ab -a ac mod n
  • b c mod n

46
properties of modulo operator Proof
  • Proof
  • Given (a x b) (a x c) mod n
  • Multiply with a-1 (Multiplicative inverse of a)
    on both sides
  • a-1 (a x b) a-1 (a x c) mod n
  • b c mod n
  • REVISION However the multiplicative inverse of
    a exists only if a and n are relatively
    prime.
  • a b mod n if n(a-b)

47
Agenda
  • After studying examples of modular arithmetic
  • Modular arithmetic
  • Divisors, GCD, Euclids theorem
  • prime numbers
  • Fields of type Zp
  • Finite Fields, Extended Euclids Theorem for
  • finding multiplicative inverse
  • Polynomial arithmetic

48
Divisors
  • If for some m, amb (a,b,m all integers),
  • that is b divides into a with no remainder ,
  • denote this as ba
  • and say that b is a divisor of a
  • eg. all of 1,2,3,4,6,8,12,24 are the divisors of
    24.

49
Properties of Divisors
  • If a1, then a ?1.
  • If ab and ba, then a ?b.
  • Any b ? 0, divides 0.
  • If bg and bh,
  • then b(mg nh)
  • for arbitrary integers m and n

50
Greatest Common Divisor
  • gcd(a,b) max k, such that ka and kb
  • Properties
  • 1. gcd is required to be positive.
  • gcd(a,b) gcd(a, -b) gcd(-a,b)
    gcd(-a,-b) gcd(a,b)
  • 2. gcd(a,0) a
  • 3. If gcd(a,b) 1, a and b are relatively prime.

51
Properties of gcd function contd
  • Assume that a b.
  • 4. gcd(a,b) gcd (b, a mod b)
  • called a Theorem on the
    next slide
  • Proof
  • let d gcd(a,b)
  • Then da and db ( i. e. a k1d and b
    k2d )
  • If (a mod b) r,
  • a kb r or r a kb
  • k1.d k. k2d
  • This proves dr.
  • Thus (4) can be repetitively used to find d.

52
Greatest Common Divisor 2 definitions
  • c gcd(a,b) is the largest number that divides
    evenly into both a and b
  • eg gcd(60,24) 12
  • Positive integer c is gcd of two positive
    integers a and b if
  • c is a divisor of a and b
  • Any divisor of a and b is a divisor of c.
  • Theorem gcd(a,b) gcd (b, a mod b)
  • RHS may be a simpler function if agtb.

53
Euclids algorithm
  • Stated in his book Elements, written in 300 BC.
    Historians believe that the algorithm was devised
    200 years earlier
  • an efficient way to find gcd(a,b)
  • derived from the observation
  • If a b have a common factor d (ie am.d
    bn.d),
  • then d is also a factor in any difference
    between them, a-p.b (m.d)-p.(n.d) d.(m-p.n).
  • uses successive instances of the theorem
  • gcd(a,b) gcd(b, a mod b)
  • Note This MUST always terminate by giving gcd
    since eventually we get a mod b 0 (no
    remainder).

54
Euclid's GCD Algorithm
  • Euclid's Algorithm to compute gcd(a,b)
  • A ? a, B ? b
  • while Bgt0
  • R A mod B
  • A ? B, B ? R
  • return A gcd(a,b)
  • The example on the next slide uses Euclids
    algorithm.
  • Even more useful Extended Euclids Algorithm
    Used for finding out the Multiplicative Inverse

55
Example GCD(1970,1066)
  • 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)
  • Hence gcd(1970,1066) 2

56
Agenda
  • After the Euclids theorem
  • Modular arithmetic
  • Divisors, GCD, Euclids theorem
  • prime numbers
  • Fields of type Zp
  • Finite Fields, Extended Euclids Theorem for
  • finding multiplicative inverse
  • Polynomial arithmetic

57
Prime Numbers
  • A prime number p an integer, whose only integer
    factors are itself and 1.
  • Aug 6, 2002 Manindra Agrawal, Neeraj Kayal,
    Nitin Saxena of IIT Kanpur
  • Theorem There is a deterministic
    polynomial-time algorithm for determining whether
    a number is a prime or a composite.
  • Odd Primes all prime numbers except 2
  • The magical prime 2, used in cryptography

58
Prime Numbers sequence Referencehttp//www.maths.
ex.ac.uk/7Emwatkins/zeta/ss-b.htm
Here the sequence of primes is presented
graphically in terms of a step function or
counting function which is traditionally denoted
as ?(x). (Note this has nothing to do with the
value 3.14159...) The height of the graph at
horizontal position x indicates the number of
primes less than or equal to x. Hence at each
prime value of x, we see a vertical jump of one
unit.
59
Prime Numbers sequence Referencehttp//www.maths.
ex.ac.uk/7Emwatkins/zeta/ss-e.htm
Now zooming out by a factor of 2500, we get the
above graph. Senior Max Planck Institute
mathematician Don Zagier, in his article "The
first 50 million primes" Mathematical
Intelligencer, 0 (1977) 1-19 states "For me,
the smoothness with which this curve climbs is
one of the most astonishing facts
in mathematics."
60
Prime Number Factors of a number
  • Unique factors of any integer a gt 1
  • a ? pap where P is the set of prime
    numbers
  • p? P and where ap is the degree
    of p
  • c a.b ? cp (apbp) for all p.
  • Ex33033 3x7x112 X13 85833 3x3x3x11x172
  • c3 31 4, c7 1, c11 2 1 3, c13 1, c17
    2
  • gcd(33033, 85833) 3x11 33
  • db ? dp ? bp for all p Thus if d 143,
    14333033
  • Calculating the prime factors of a large number
    is a difficult task. So prime number
    factorization ? NOT used for evaluation of a.b or
    of the greatest common divisor (gcd) of a and b.

61
Agenda
  • After discussing prime numbers
  • Modular arithmetic
  • Divisors, GCD, Euclids theorem
  • prime numbers
  • Fields of type Zp
  • Finite Fields, Extended Euclids Theorem for
  • finding multiplicative inverse
  • Polynomial arithmetic
  • with coefficient obeying modulo n arithmetic
  • with modulo m(x) and with coefficient
    obeying
  • modulo n arithmetic

62
Modular Arithmetic
  • Consider the set of non negative integers
  • Zp 0, 1, 2, 3(p-1)
  • Each element of Zp represents a residue class
    modulo p where p is a prime number.
  • Properties of Modular Arithmetic for Integers in
    Zp are given in table 4.2 (Stallings) 4th Ed.

63
Table 4.2 Reference Page
105 Stallings, 4th Edition
Properties Expressions
Commutative Laws (wx) mod p (xw) mod p (w.x) mod p (x.w) mod p
Associative laws (wx) y mod p w(xy) mod p (w.x). y mod p w.(x.y) mod p
Distributive Laws w. (x y) mod p w.x w.y mod p
Identities (0 w)mod p w mod p (1 . w) mod p w mod p
Additive inverse (-w) Multiplicative Inverse (w-1) For each w ? Zp , there exists a z such that wz ? 0 mod n For each w ? Zp ,there exists a z such that w .z 1 mod p
64
Agenda
  • After discussing Fields of type Zp
  • Modular arithmetic
  • Divisors, GCD, Euclids theorem
  • prime numbers
  • Fields of type Zp
  • Finite Fields, Extended Euclids Theorem for
  • finding multiplicative inverse
  • Polynomial arithmetic

65
Order of a Finite Field
  • Order of a Finite Field the number of elements
    in the field
  • For
  • Zp 0, 1, 2, 3(p-1)
  • Order p

66
Galois Fields
  • Galois Field GF(pn) A finite field of order pn
  • For p any prime integer and
  • n any integer, greater than or equal to 1, there
    is a unique field with pn elements, denoted by
    GF(pn).
  • Unique Any two fields with the same number of
    elements must be essentially the same, except
    perhaps for giving the elements of the field
    different names. ? An interesting fact

67
Galois fields of interest in cryptography
  • GF(p)
  • GF(2n).
  • Let us first consider GF(p)
  • GF(p) 0, 1, 2, . (p-1), with arithmetic
    operations modulo p.

68
Galois Fields GF(p)
Some Properties
  • Every element in GF(p) relatively prime to p
  • ? every element has a multiplicative inverse.
  • ? Hence GF(p) is a Field.
  • CHARACTERISTIC of a Field The number of times a
    multiplicative identity can be added to itself
    before you get to zero.
  • For GF(p), Characteristic the number of
    elements in the field p.
  • A Field of characteristic p Fp

69
Mutiplicative Inverse Algorithm
  • finding the multiplicative inverse of b, such
    that b.b-1 1
  • Given that b ltm
  • Extended Euclid (m,b) Algorithm
  • To find c such that c.b 1 mod m

70
Finding Inverses
for mgtgtb
  • EXTENDED EUCLID(m, b) ALGORITHM
  • 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 b1 mod m
  • i.e. B2 multiplicative inverse of b
  • 4. Q ?A3/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

71
Example Inverse of 550 in GF(1759)
Ti Ai Bi x Q
Hence 355 is multiplicative inverse of 550 mod
1759. If B2 be ve, subtract it from m to get the
answer.
72
Finite Field GF(2)
  • A B AB A-B A.B
  • 0 0 0 0 0
  • 0 1 1 1 0
  • 1 0 1 1 0
  • 1 1 0 0 1
  • Thus in GF(2),
  • ab a-b is an XOR operation.
  • a.b is an AND operation.

73
Agenda
  • Polynomial arithmetic
  • (Ordinary polynomial algebra is of no interest in
    cryptography.)
  • with coefficients obeying modulo n arithmetic
  • Prime polynomials and polynomial gcd
  • with modulo m(x) and with coefficient obeying
  • modulo n arithmetic

74
Polynomial Arithmetic
  • Consider a polynomial
  • A zero-th degree polynomial is a constant
    polynomial.
  • A nth degree polynomial is called a MONIC
    polynomial, if an 1.
  • several alternatives available
  • ordinary polynomial arithmetic Not used in
    cryptography
  • poly arithmetic with coeff arithmetic as mod p
    called polynomial basis over a finite field
  • poly arithmetic with coeff mod p and polynomials
    mod M(x)

75
A Revision Group, Ring and Field
A1 closure under addition
A2 Associativity of addition
A3 Additive identity
A4 Additive inverse
Group
Abelian Group
A5 Commutativity of addition
M1 closure under multiplication
M2 Associativity of multiplication
Ring
M3 Distributive laws
M4 Commutativity of multiplication
Commutative Ring
M5 Multiplicative identity
Integral domain
M6 No zero divisors
Field
M7 Multiplicative inverse
76
Polynomial Arithmetic with
Modulo Coefficients
  • Poly arithmetic is based on the fact that powers
    of x are linearly independent
  • Let coefficients be elements of a Field GF(p).
  • The set of such polynomials forms a polynomial
    ring.
  • Difference between a Field and a Ring Consider
    two elements a and b.
  • Field a/b a.b-1 is also an element of the
    field.
  • Ring (that is not a Field) b-1 may not exist
    as an element of the Ring. ( a/b may not result
    in an exact division.)
  • Even if the coeff are the elements of a Field,
    the division of polynomials may leave a
    remainder.

77
Polynomials over GF(2)
  • In cryptography, we are interested in mod 2
  • all coefficients are 0 or 1
  • The coeff use modulo 2 arithmetic
  • EXAMPLE f(x) x3 x2 and g(x) x2 x 1
  • ADDITION f(x) g(x) x3 x 1
  • Addition of polynomials requires XOR of
    coeffs
  • MULTIPLICATION
  • multiplication of g(x) with x3 x5 x4 x3
  • multiplication of g(x) with x2 x4 x3 x2
  • f(x) . g(x) x5 x2

78
Polynomials over GF(2)
Multiplication and Addition
  • f(x) 1100
  • g(x)0111
  • Addition XOR process yields 1011
  • Multiplication Uses shifting and XOR
  • multiplication of g(x) with x3 111000 Lshift by
    3
  • multiplication of g(x) with x2 011100 Lshift by
    2
  • f(x) . g(x) 100100

79
Agenda
  • Polynomial arithmetic
  • (Ordinary polynomial algebra is of no interest in
    cryptography.)
  • with coefficients obeying modulo n arithmetic
  • Prime polynomials and polynomial gcd
  • with modulo m(x) and with coefficient obeying
    modulo n arithmetic

80
Modulo m(x) A preliminary view
  • Multiplication increases the degree of the
    resultant polynomial.
  • To ensure that the degree remains the same, we
    may consider
  • ( f(x) . g(x) ) mod m(x).
  • If a(x) f(x) . g(x),
  • a(x) q(x).m(x) r(x),
  • ( f(x) . g(x) ) mod m(x) may be said to be equal
    to r(x)
  • The degree of r(x) lt that of m(x).

81
A Prime Polynomial
  • can write any polynomial in the form
  • a(x) q(x) m(x) r(x)
  • if the remainder is zero, m(x) divides a(x)
  • If f(x), over a Field F, has no divisors other
    than itself 1, it is called
  • an irreducible (or prime) polynomial.
  • Another definition f(x), over a Field F, is
    irreducible, iff f(x) cannot be expressed as a
    product of two
  • polynomials, both of degree lower than that of
    f(x).

82
Polynomial GCD
  • Definition c(x) is the greatest common divisor
    of a(x) and b(x) if
  • c(x) divides both a(x) and b(x).
  • Any divisor of a(x) and b(x) is a divisor of
    c(x).
  • Euclids Algorithm to find polynomial gcd
  • Based on
  • gcda(x), b(x) gcdb(x), a(x) mod b(x)
  • with the assumption that
  • the degree of a(x) gt the degree of b(x).

83
Euclids Algorithm to find gcda(x), b(x)
-- similar to Extended Euclid(m, b) Algorithm
  • gcda(x), b(x)
  • Assume the degree of a(x) gt the degree of b(x).
  • 1. A(x) ? a(x) B(x) ? b(x)
  • 2. if B(x) 0 return A(x) gcda(x), b(x)
  • 3. R(x) A(x) mod B(x)
  • 4. A(x) ? B(x)
  • 5. B(x) ? R(x)
  • 6. goto 2

84
Euclids Algorithm to find gcda(x), b(x)
An Example
  • Givena(x) x6x5x4x3x2x1
  • b(X) x4 x2 x1
  • Euclids Algorithm
  • A x6x5x4x3x2x1x1 x4 x2 x1 x3 x21
  • B x4 x2 x1 x3 x21 0
  • R x3 x21 0
  • Q x2 x x 1
  • gcda(x), b(x) A(x) x3 x21

85
Agenda
  • Polynomial arithmetic
  • (Ordinary polynomial algebra is of no interest in
    cryptography.)
  • with coefficients obeying modulo n arithmetic
  • Prime polynomials and polynomial gcd
  • with modulo m(x) and with coefficient obeying
  • modulo n arithmetic

86
Polynomials over GF(2)
  • Polynomial arithmetic modulo an irreducible
    polynomial forms a Field.
  • By analogy with modulo operations studied
    earlier, if a and b are relatively prime, the
    multiplicative inverse exists.
  • We shall look at an extended Euclid algorithm
    to evaluate the multiplicative inverse of a(x)
    modulo b(x), where b(x) is an irreducible
    polynomial.
  • On the coefficients, the arithmetic is modulo 2.

87
Extracts from earlier slides
  • If a mod 7 b mod 7, a and b are said to be
    congruent mod 7.
  • O .,-21,-14,-7,0,7,14.
  • is called a Residue Class Mod 7.
  • The Smallest Non-negative integer of the class is
    used to represent the class.
  • To find the smallest Non-negative integer, to
    which k is congruent, is called reducing k modulo
    n
  • Zp 0, 1, 2, 3(p-1)
  • Each element of Zp represents a residue class
    modulo p where p is a prime number.

88
Set of Residues modulo m(x)
  • m(x) nth degree polynomial
  • Example residue class (x1), modulo m(x)
    consists of all such polynomials a(x) such that
  • a(x) (x1)mod m(x)
  • Or all the polynomials, which satisfy
  • a(x) mod m(x) x1.
  • For m(X) x3 x1,
  • one possible value of a(x) is x4 x2 1.

89
GF (pn) with an irreducible polynomial b(x)
  • Set of residues
  • consisting of pn elements.
  • Each of these elements represented by one of the
    pn polynomials of degree mltn
  • Example GF (23)
  • with an irreducible polynomial b(x) x3 x1
  • The set of residues are
  • 0, 1, x, (x1), x2, (x2 1), (x2 x), (x2x1)
  • Finding Multiplicative inverse of b(x) modulo
    m(x)
  • Assume degree of b(x) lt degree of m(x)
  • gcdm(x),b(x) 1

90
23 elements of finite polynomial
field GF(23)
  • Decimal number Binary number Polynomial
  • 0 000 0
  • 1 001 1
  • 2 010 x
  • 3 011 x1
  • 4 100 x2
  • 5 101 x21
  • 6 110 x2x
  • 7 111 x2x1
  • Choose m(x)(x3x1) as the irreducible
    polynomial.

91
Example GF(23)
92
Multiplicative Inverse a(x).b(x) mod (x3 x1)
1
a(x) b(x) a-1(x)
x x2 1
x 1 x2 x
x2 x2 x 1
x2 1 x
x2 x x 1
x2 x 1 x2
1 1
93
Additive and Multiplicative Inverses in GF (23)
  • w 0 1 2 3 4 5 6 7
  • Additive Inverse
  • -w 0 1 2 3 4 5 6 7
  • Multiplicative Inverse
  • w-1 1 5 6 7 2 3 4
  • If mult results in a polynomial a(x) of degree
    greater than 2 (ie n-1 for pn or a degree
    greater than or equal to n), reduce it to a
    polynomial, r(x), of degree less than or equal to
    2 by using
  • r(x) a(x)
    mod(x3x1).

94
Multiplicative inverse Extended
Euclidm(x), b(x) Algorithm
  • (A1, A2, A3)? (1, 0, m)
  • (B1, B2, B3)? (0, 1, b)
  • If B3 0,
  • return A3 gcd(m, b) no inverse
  • If B3 1
  • return B2 as the multiplicative inverse of B
  • (i.e. b(x).B2 1 mod m(x) )
  • Q ?A3/B3?
  • (T1, T2, T3)? (A1 Q B1, A2 Q B2, A3 QB3)
  • (A1, A2, A3)? (B1, B2, B3)
  • (B1, B2, B3)? (T1, T2, T3)
  • Go to 2

95
Modular Polynomial Arithmetic
  • can compute in field GF(2n)
  • polynomials with coefficients modulo 2
  • The elements of GF are polynomials, whose degree
    is less than n
  • hence must reduce modulo an irreducible poly of
    degree n (for multiplication only)
  • The polynomials form a finite field. The number
    of elements in the field is 2n.
  • For every element of the field, a multiplicative
    inverse can always be found by using Euclids
    Inverse algorithm.

96
ARITHMETIC OPERATIONS GF(28) with m(x)
(x8x4x3x1)
  • AES uses GF(28) and an irreducible polynomial
    (x8x4x3x1).
  • In binary, it is 100011011
  • In HEX, the polynomial 0x11B
  • Justification The first out of the 30
    irreducible polynomials of degree 8, given in
    Lidl, R., Niederreiter, H. Introduction to
    Finite Fields and Their Applications, Cambridge
    University Press, 1994
  • For comments on how to choose a prime polynomial
    for a specific size of the field, please see the
    paper by E. De Win et al.
  • Reference E. De Win, A. Bosselaers, S.
    Vandenberghe, P. De Gersem and J.VandeWalle, A
    fast Software Implementationfor Arithmetic
    Operations in GF(2n) , ASIACRYPT 96,
    Springer-Verlag, pp 65-76

97
MULTIPLICATIVE INVERSE To find c(x) such that
(x7x1).c(x) 1 mod(x8x4x3x1)
  • A1 1 0 1
    x3 x21
  • A2 0 1 x
    x4x3 x1
  • A3 x8x4x3x1 x7x1 x4x3 x21 x
  • B1 0 1 x3 x21
    x6x2 x1
  • B2 1 x x4x3 x1
    x7
  • B3 x7x1 x4x3 x21 x
    1
  • Q - x x3 x21
    x3 x2x
  • Answer The Multiplicative Inverse of
  • (x7x1) mod(x8x4x3x1) c(x) x7

98
  • "Genius is condemned by a malicious social
    organization to an eternal denial of justice in
    favor of fawning mediocrity"
  • -- Evariste Galois

99
Representation
  • A polynomial with coeff, obeying modulo 2
    arithmetic, can be represented by a binary or a
    HEX number.
  • Example 0x11B 100011011 represents

  • x8x4x3x1.
  • This is an irreducible polynomial.
  • A polynomial in GF (28), a(x)
    a7x7a6x6a1xa0
  • can be represented as ( a7 a6 a5. a1 a0 )
  • Addition of two polynomials a(x) and b(x) Use
    XOR operation on two bit arrays
  • ( a7 a6 a5.. a1 a0 ) ? ( b7 b6 b5 ..b1 b0 )

100
ARITHMETIC OPERATIONS MULTIPLICATION for
GF(28) with m(x) (x8x4x3x1)
  • Reduction
  • Example 1
  • x8 mod m (x) m (x) x8 x4 x3 x 1
  • Note x4 x3 x 1 can be represented as
    0x1B.
  • In general xn mod m (x) m (x) xn
  • Multiplication Let b(x) b7x7 b6x6 b1x
    b0
  • Example 2 Consider multiplication of b (x) with
    x
  • x . b (x) mod m (x)
  • if b7 0, x b (x) is in the reduced form.
  • If b7 1 using results of Example 1,
  • (b6x7b1x2b0x) ? (x4 x3 x 1)

101
ARITHMETIC OPERATIONS MULTIPLICATION
Generalized Result
  • This multiplication x . b (x) mod m (x) is done
    as follows
  • x . b (x) mod m (x) b6b5b4b3b2b1b00 if b7
    0
  • (b6b5b4b3b2b1b00) ? (00011011) if b7 1
  • Multiplication by a higher power can be achieved
    by a repeated application of Step2.
  • Example 3
  • r (x) b (x) . a (x) mod m (x)
  • (x6 x4 x2 x 1) . (x7 x 1) mod
    (x8x4 x3 x 1)

102
ARITHMETIC OPERATIONS MULTIPLICATION
Example 3
  • To get r (x),
  • Step1
  • (x6x4 x2 x 1) . x mod m (x)
  • (0101 0111) . (0000 0010)
  • Shift left ? 1010 1110
  • step2
  • (x6x4 x2 x 1) . x2 mod m (x)
  • (0101 0111) . (0000 0100)
  • (1010 1110) . (0000 0010) ? ( 0001 1011)
  • (0101 1100) ? (0001 1011)
  • (0100 0111)

103
ARITHMETIC OPERATIONS MULTIPLICATION Example
(continued)
  • Step3
  • (x6 x4 x2 x 1) . x3 mod m (x)
  • (0101 0111) . (0000 1000)
  • (0100 0111) . (0000 0010)
  • 1000 1110
  • Step4 Multiplication of b (x) by x4 mod m (x)
  • (0101 0111) . (0001 0000)
  • (1000 1110) . (0000 0010) ? (0001 1011)
  • (0001 1100) ? (0001 1011)
  • (0000 0111)

104
ARITHMETIC OPERATIONS MULTIPLICATION Example
(continued)
  • Step5 Multiplication of b (x) by x5 mod m (x)
  • (0101 0111) . (0010 0000)
  • (0000 0111) . (0000 0010)
  • 0000 1110
  • Step6 Multiplication of b (x) by x6 mod m (x)
  • Result 0001 1100
  • Step7 Multiplication of b (x) by x7 mod m (x)
  • Result 0011 1000

105
ARITHMETIC OPERATIONS MULTIPLICATION Example
(continued)
  • Step8 b (x) . a (x) mod m (x) where a (x) x7
    x 1
  • (0011 1000) ? (1010 1110) ? ( 0101 0111)
  • 1100 0001
  • Hence
  • b (x) . a (x) mod m (x)
  • (x6x4 x2 x 1) . (x7 x 1) mod (x8x4
    x3 x 1)
  • x7x6 1

106
Computational Considerations
  • Since coefficients are 0 or 1, any such
    polynomial can be represented as a bit string.
  • Addition becomes XOR of the bit strings.
  • Multiplication is shift or shift XOR.
  • cf long-hand multiplication
  • See, again, the line in red, five slides back.
  • Modulo reduction done by repeatedly applying the
    rule of that slide.

107
Use of the bit notation for polynomials Ex for
GF(28) with m(x) x8x4x3x1.
  • Example rc1(x) 1
  • rcj(x) x.rcj-1(x) mod m(x) for j 2 to 10
  • Denoted by RC(1) 1
  • RC(j) 2.RC(j-1) for j 2 to 10
  • For GF(28), the number of members of the finite
    group are 256, starting from 0 to 255.
  • Thus RC(2) 2,RC(8) 128
  • rc9(x) x8 mod m(x) x4x3x1 ? RC(9) 1B
  • RC(10) 0011 0110 3616 x5x4x2x
  • obtained by shifting RC(9) to
    the left

108
Win thousands of dollars!
  • Solve problems in Number theory, Graph theory and
    Combinatorics-- and WIN!
  • Paul Erdos, the great Hungarian problem solver,
  • is the purser of all of the problems.
  • (The purser is the final judge and arbiter of
    prize-winning solutions.
  • The award only goes to the person who solves a
    problem first, and
  • the purser is the arbiter of that too.)
  • Volunteer Advisor for solvers greg_at_math.berkeley.
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  • References 1.A Tribute to Paul Erdos,
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  • Press, 1990, pp. 467-477. 2. Paths, Flows, and
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  • Springer-Verlag, 1980, pp. 35-45. 3. Erdos on
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  • of unsolved problems, Fan Chung RonGraham, AK
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