S is a ring. It satisfies the properties of a ring:
- It is closed under addition and multiplication.
- Addition and multiplication are both associative.
- Addition has an identity element and inverse elements.
- Addition is commutative.
- Multiplication distributes over addition.
–concept of groups, rings, fields
–modular arithmetic with integers
–Euclid’s algorithm for GCD
–finite fields GF(p)
–polynomial arithmetic in general and in GF(2n)
–concept of groups, rings, fields
–modular arithmetic with integers
–Euclid’s algorithm for GCD
–finite fields GF(p)
–polynomial arithmetic in general and in GF(2n)
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
This PowerPoint was created to help out graduating seniors who are taking the TAKS Mathematics Exit-Level test. It includes formulas, rules & things that they need to remember to pass the test.
A study on number theory and its applicationsItishree Dash
A STUDY ON NUMBER THEORY AND ITS APPLICATIONS
Applications
Modular Arithmetic
Congruence and Pseudorandom Number
Congruence and CRT(Chinese Remainder Theorem)
Congruence and Cryptography
This PowerPoint was created to help out graduating seniors who are taking the TAKS Mathematics Exit-Level test. It includes formulas, rules & things that they need to remember to pass the test.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Acetabularia Information For Class 9 .docxvaibhavrinwa19
Acetabularia acetabulum is a single-celled green alga that in its vegetative state is morphologically differentiated into a basal rhizoid and an axially elongated stalk, which bears whorls of branching hairs. The single diploid nucleus resides in the rhizoid.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
2. Introduction
• finite fields has 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
3. Divisors
• We say that a non-zero number b divides a if
for some m have a=mb (a,b,m all integers)
• that is b divides 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
4. 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 , 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
5. Division Algorithm
• Given any positive integer n and any nonnegative
integer a, if we divide a by n, we get an integer
quotient q and an integer remainder r that obey
the following relationship:
• a = qn + r where 0 <= r < n; q = floor(a/n)
• remainder r often referred to as a residue
6. 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
7. 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)
9. 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
• For a given positive integer , two integers a and b are
called congruent modulo , written a ≡ b (mod n)
• if a-b is divisible by n (or equivalently if a and b have
the same remainder when divided by n ).
• e.g., 37 ≡ 57 (mod 10)
10. 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
11. 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
15. 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);
16. Extended Euclidean Algorithm
• For given integers a and b, the extended Euclidean algorithm not
only calculates the greatest common divisor d but also two
additional integers x and y that satisfy the following equation
• 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 the end find GCD value and also x & y
• The Bézout coefficients are x and y, that is GCD(a,b)=r=ax+by
• if GCD(a,b)=1 these values are inverses
17. Example
• The following table shows how the extended Euclidean
algorithm proceeds with input 240 and 46.
• The greatest common divisor is the last non zero
entry, 2 in the column "remainder".
• The computation stops at row 6, because the
remainder in it is 0.
• Bézout coefficients appear in the last two entries of the
second-to-last row.
• In fact, it is easy to verify that -9 × 240 + 47 × 46 = 2.
• Finally the last two entries 23 and -120 of the last row
are, up to the sign, the quotients of the
input 46 and240 by the greatest common divisor 2.
23. Finding Inverse
Problem: Find an inverse of a (mod n) for the pair
of relatively prime numbers a=4, n=9
• gcd(4,9)= gcd(9,4)
• 9=4x2 + 1
• 4=1x4+0
Gcd(9,4)=mx9+ nx4
1=9 – 4x2 =1.9+-2x4
Bezut’s coeffiecients are(1,-2)
Inverse of 4 (mod 9)= -2 (mod 9)= 7 (mod 9)
So, 7 is the required inverse answer.
24. Example
Problem: Find an inverse of a (mod n) for the pair
of relatively prime numbers a=19, n=141
• Gcd(19,141)= gcd(141,19)
• 141=19.7 + 8
• 19=8.2+3
• 8=3.2+2
• 3=2.1 +1
• 2=1.2+0
• So,gcd(19,141)=1
26. • Problem: Find the multiplicative inverse of 8 mod 11, using
the Euclidean Algorithm.
• Sol:
11 = 8(1) + 3 3 = 11 − 8(1) 8 = 3(2) + 2 2 = 8 − 3(2) 3 = 2(1) + 1 1
= 3 − 2(1) 2 = 1(2)
Now reverse the process using the equations on the right.
1 = 3 − 2(1) 1 = 3 − (8 − 3(2))(1) = 3 − (8 − (3(2)) = 3(3) − 8 1 = (11
− 8(1))(3) − 8 = 11(3) − 8(4) = 11(3) + 8(−4)
Therefore 1 ≡ 8(−4) mod 11, or if we prefer a residue value for
the multiplicative inverse, 1 ≡ 8(7) mod 11.
27. Problem: Find the multiplicative inverse of 17 mod 43, using the
Euclidean Algorithm.
Sol: 17x ≡ 1 (mod 43) So, x ≡ 17-1 (mod 43)
43= 17 x2 +9 17= 9 x1 +8 9= 8 x1 + 1
1= 9 – 8
1=9 – (17-9) [using 17= 9x1 + 8]
=2.9 – 17
= 2(43- 17x2) – 17[ using 43= 17x2 + 9]
=2x43 – 5x17
17-1 (mod 43) ≡ -5 ≡ 38
x=38
28. Problem: Find the Multiplicative Inverse of 51
mod 88
Inverse of 51 mod 88 is 19. 51*19 mod 88 = 1
29. Group
• a set of elements or “numbers”
– may be finite or infinite
• with some operation whose result is also in the set (closure)
• group is a set of elements that is closed under a binary
operation and that is associative and that includes an
identity element and an inverse element
• 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
– EXAMPLE???? set of integers with addition
30. 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
• Example: {1, 2, 4, 8} with mod 12 multiplication, the
generator is 2.
31. Ring
• A ring is a set of elements that is closed under two binary
operations, addition and subtraction, with the following: the
addition operation is a group that is commutative; the
multiplication operation is associative and is distributive
over the addition operation.
• 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
• 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
32. Consider the set S = {a, b} with addition and multiplication defined by the
tables:
Is S a ring? Justify your answer.
Solution:
(A1) Closure: The sum of any two elements in S is also in S.
(A2) Associative: S is associative under addition, by observation.
(A3) Identity element: a is the additive identity element for
addition.
(A4) Inverse element: The additive inverses of a and b are b and
a, respectively.
(A5) Commutative: S is commutative under addition, by
observation.
(M1) Closure: The product of any two elements in S is also in S.
(M2) Associative: S is associative under multiplication, by
observation.
(M3) Distributive laws: S is distributive with respect to the two
operations, by observation.
33. Field
A field is a nonzero commutative ring that contains
a multiplicative inverse for every nonzero element, or
equivalently a ring whose nonzero elements form
an abelian group under multiplication.e.g., the real
numbers R w.r.t the usual operation + & *.
two operations which form:
abelian group for addition
abelian group for multiplication (ignoring 0)
ring
have hierarchy with more axioms/laws
group -> ring -> field
35. 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)
36. 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)
38. 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)
39. 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
40. 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
41. 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
42. 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));
43.
44. Problem: Find the GCD of 3x6 + 2x2+x +5 and 6x4 + x3 + 2x +4 in
F7[x]
Solution: Divide
3x6 + 2x2+x +5 by 6x4 + x3 + 2x +4
Q: 4x2+ 4x +4 R: 2x3 +6x2+5x +3
Divide
3x6 + 2x2+x +5 by 2x3 +6x2+5x +3
Q: 3x +2 R: x2+4x+5
Divide 2x3 +6x2+5x +3 by x2+4x+5
Q: 2x +5 R: 3x +6
Divide x2+4x+5 by 3x +6
Q: 5x +3 R: 1
45. Problem: determine the GCD for the pair of
polynomials X3 + x +1 and x2 + x +1 over GF(2).
Ans: 1
46. 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
50. 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)
51. 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
52. 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