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                           <rdf:li><rdf:Description xmpDM:name="Overview The Euclidean Algorithm for GCD Modular Arithmetic Groups, Rings, and Fields Galois Fields GF(p) Polynomial Arithmetic These slides are" xmpDM:startTime="17300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="partly based on Lawrie Brown’s slides supplied with William Stalling’s book “Cryptography and Network Security: Principles and Practice,” 7th Ed," xmpDM:startTime="17300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="2017." xmpDM:startTime="17300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="17300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euclid's Algorithm Goal: To find greatest common divisor Example: gcd(10,25)=5 using long division 10) 25 (2 20 -- 5)10 (2" xmpDM:startTime="40733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="10 -- 00 Test: What is GCD of 12 and 105?" xmpDM:startTime="40733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euclid's Algorithm: Tabular Method ri = ui x + vi y ui = ui-2 - qi ui-1 vi = vi-2" xmpDM:startTime="170033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="- qi vi-1 Finally, If ri = 0, gcd(x,y) = ri-1 Write the first 2 rows." xmpDM:startTime="170033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Set i = 2." xmpDM:startTime="170033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Divide ri-1 by ri, write quotient qi+1 on the next row Fill out the remaining entries in the new bottom" xmpDM:startTime="170033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="row: Multiply ri by qi+1 and subtract from ri-1 Multiply ui by qi+1 and subtract from ui-1 Multiply vi by" xmpDM:startTime="170033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="qi+1 and subtract from previous vi-1 x y" xmpDM:startTime="170033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euclid’s Algorithm Tabular Method (Cont) Example 2: Fill in the blanks" xmpDM:startTime="573900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Tabular Method (Cont) Example 2: Fill in the blanks 8 1 5 q i r i u i v i" xmpDM:startTime="619833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="15 0 1 8 1 0 1 7 - 1 1 1 1 2 - 1 7 0 - 15" xmpDM:startTime="619833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="- 8 GCD(8,15) = 1 ri = ui x + vi y If gcd(x, y) = 1, 1=2*8-1*15 or 2*8" xmpDM:startTime="619833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 1+1*15 2*8 mod 15 = (1+1*15)mod 15 = 1 Þ Inverse of 8 = 2 mod 15 In general," xmpDM:startTime="619833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ui x + vi y = 1 Þ x-1 mod y = ui Þ ui is the inverse of x" xmpDM:startTime="619833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="in “mod y” arithmetic." xmpDM:startTime="619833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="619833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4A Find the multiplicative inverse of 5678 mod 8765 Do it on your own." xmpDM:startTime="1038633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Do not submit." xmpDM:startTime="1038633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Answer: 2527" xmpDM:startTime="1038633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Tabular Method (Cont) Example 2: Fill in the blanks 8 1 5 q i r i u i v i" xmpDM:startTime="1076400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="15 0 1 8 1 0 1 7 - 1 1 1 1 2 - 1 7 0 - 15" xmpDM:startTime="1076400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="- 8 GCD(8,15) = 1 ri = ui x + vi y If gcd(x, y) = 1, 1=2*8-1*15 or 2*8" xmpDM:startTime="1076400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 1+1*15 2*8 mod 15 = (1+1*15)mod 15 = 1 Þ Inverse of 8 = 2 mod 15 In general," xmpDM:startTime="1076400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ui x + vi y = 1 Þ x-1 mod y = ui Þ ui is the inverse of x" xmpDM:startTime="1076400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="in “mod y” arithmetic." xmpDM:startTime="1076400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1076400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4A Find the multiplicative inverse of 5678 mod 8765 Do it on your own." xmpDM:startTime="1216800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Do not submit." xmpDM:startTime="1216800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Answer: 2527" xmpDM:startTime="1216800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Modular Arithmetic xy mod m = (x mod m) (y mod m) mod m (x+y) mod m = ((x mod" xmpDM:startTime="1271467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="m)+ (y mod m)) mod m (x-y) mod m = ((x mod m)- (y mod m)) mod m x4 mod" xmpDM:startTime="1271467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="m = (x2 mod m)(x2 mod m) mod m xij mod m = (xi mod m)j mod m 125 mod" xmpDM:startTime="1271467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="187 = 125 (225+285) mod 187 = (225 mod 187) + (285 mod 187) = 38+98 = 136 1252 mod" xmpDM:startTime="1271467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="187 = 15625 mod 187 = 104 1254 mod 187 = (1252 mod 187)2 mod 187 = 1042 mod 187" xmpDM:startTime="1271467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10816 mod 187 = 157 1256 mod 187 = 1254+2 mod 187 = (157104) mod 187 = 59" xmpDM:startTime="1271467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Modular Arithmetic Operations Z = Set of all integers = {…, -2, -1, 0, 1, 2, …} Zn = Set" xmpDM:startTime="1510500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="of all non-negative integers less than n = {0, 1, 2, …, n-1} Z2 = {0, 1} Z8 = {" xmpDM:startTime="1510500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0, 1, 2, 3, 4, 5, 6, 7} Addition, Subtraction, Multiplication, and division can all be defined in Zn For" xmpDM:startTime="1510500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Example: (5+7) mod 8 = 4 (4-5) mod 8 = 7 (57) mod 8 = 3 (3/7) mod 8 =" xmpDM:startTime="1510500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="5 (5*5) mod 8 = 1" xmpDM:startTime="1510500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Modular Arithmetic xy mod m = (x mod m) (y mod m) mod m (x+y) mod m = ((x mod" xmpDM:startTime="1561000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="m)+ (y mod m)) mod m (x-y) mod m = ((x mod m)- (y mod m)) mod m x4 mod" xmpDM:startTime="1561000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="m = (x2 mod m)(x2 mod m) mod m xij mod m = (xi mod m)j mod m 125 mod" xmpDM:startTime="1561000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="187 = 125 (225+285) mod 187 = (225 mod 187) + (285 mod 187) = 38+98 = 136 1252 mod" xmpDM:startTime="1561000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="187 = 15625 mod 187 = 104 1254 mod 187 = (1252 mod 187)2 mod 187 = 1042 mod 187" xmpDM:startTime="1561000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10816 mod 187 = 157 1256 mod 187 = 1254+2 mod 187 = (157104) mod 187 = 59" xmpDM:startTime="1561000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Modular Arithmetic Operations Z = Set of all integers = {…, -2, -1, 0, 1, 2, …} Zn = Set" xmpDM:startTime="1573533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="of all non-negative integers less than n = {0, 1, 2, …, n-1} Z2 = {0, 1} Z8 = {" xmpDM:startTime="1573533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0, 1, 2, 3, 4, 5, 6, 7} Addition, Subtraction, Multiplication, and division can all be defined in Zn For" xmpDM:startTime="1573533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Example: (5+7) mod 8 = 4 (4-5) mod 8 = 7 (57) mod 8 = 3 (3/7) mod 8 =" xmpDM:startTime="1573533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="5 (5*5) mod 8 = 1" xmpDM:startTime="1573533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Modular Arithmetic Properties" xmpDM:startTime="1740033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4B Determine 125107 mod 187 Do it on your own." xmpDM:startTime="1904800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Do not submit." xmpDM:startTime="1904800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Answer: 5" xmpDM:startTime="1904800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Group Group: A set of elements that is closed with respect to some operation." xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Closed Þ The result of the operation is also in the set The operation obeys: Obeys associative law:(a.b).c" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= a.(b.c)" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Has identity e: e.a" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= a.e" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= a Has inverses a-1: a.a-1" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= e Abelian Group: The operation is commutative a.b" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= b.a" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Example: Z8, + modular addition, identity =0" xmpDM:startTime="2080767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Cyclic Group Exponentiation: Repeated application of operator example: a3 = a.a.a" xmpDM:startTime="2209967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Cyclic Group: Every element is a power of some fixed element, i.e.," xmpDM:startTime="2209967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="b = ak for some a and every b in group a is said to be a generator of the" xmpDM:startTime="2209967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="group Example: {1, 2, 4, 8} with mod 12 multiplication, the generator is 2." xmpDM:startTime="2209967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="20=1, 21=2, 22=4, 23=8, 24=4, 25=8" xmpDM:startTime="2209967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Ring Ring: A group with two operations: addition and multiplication The group is Abelian with respect to addition: a+b =" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="b+a Multiplication and additions are both associative: a+(b+c)=(a+b)+c a.(b.c)=(a.b).c" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Multiplication distributes over addition a.(b+c)=a.b+a.c" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(a+b).c" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= a.c" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ b.c" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Commutative Ring: Multiplication is commutative, i.e.," xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="a.b" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= b.a" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Integral Domain: multiplication operation has an identity and no zero divisors Ref: http://en.wikipedia.org/wiki/Ring_%28mathematics%29" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2299267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4C Consider the set S = {a, b, c} with addition and multiplication defined by the following tables: Is" xmpDM:startTime="2411500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="S a ring? Justify your answer." xmpDM:startTime="2411500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2411500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Ring Ring: A group with two operations: addition and multiplication The group is Abelian with respect to addition: a+b =" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="b+a Multiplication and additions are both associative: a+(b+c)=(a+b)+c a.(b.c)=(a.b).c" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Multiplication distributes over addition a.(b+c)=a.b+a.c" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(a+b).c" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= a.c" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ b.c" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Commutative Ring: Multiplication is commutative, i.e.," xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="a.b" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= b.a" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Integral Domain: multiplication operation has an identity and no zero divisors Ref: http://en.wikipedia.org/wiki/Ring_%28mathematics%29" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2463100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4C Consider the set S = {a, b, c} with addition and multiplication defined by the following tables: Is" xmpDM:startTime="2473400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="S a ring? Justify your answer." xmpDM:startTime="2473400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2473400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Field Field: An integral domain in which each element has a multiplicative inverse." xmpDM:startTime="2493200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2493200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4C Consider the set S = {a, b, c} with addition and multiplication defined by the following tables: Is" xmpDM:startTime="2494833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="S a ring? Justify your answer." xmpDM:startTime="2494833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2494833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Ring Ring: A group with two operations: addition and multiplication The group is Abelian with respect to addition: a+b =" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="b+a Multiplication and additions are both associative: a+(b+c)=(a+b)+c a.(b.c)=(a.b).c" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Multiplication distributes over addition a.(b+c)=a.b+a.c" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(a+b).c" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= a.c" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ b.c" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Commutative Ring: Multiplication is commutative, i.e.," xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="a.b" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= b.a" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Integral Domain: multiplication operation has an identity and no zero divisors Ref: http://en.wikipedia.org/wiki/Ring_%28mathematics%29" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2495100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4C Consider the set S = {a, b, c} with addition and multiplication defined by the following tables: Is" xmpDM:startTime="2510700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="S a ring? Justify your answer." xmpDM:startTime="2510700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2510700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Field Field: An integral domain in which each element has a multiplicative inverse." xmpDM:startTime="2562933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2562933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Fields or Galois Fields Finite Field: A field with finite number of elements Also known as Galois Field The" xmpDM:startTime="2662233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="number of elements is always a power of a prime number." xmpDM:startTime="2662233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Hence, denoted as GF(pn) GF(p) is the set of integers {0,1, … , p-1} with arithmetic operations modulo prime p" xmpDM:startTime="2662233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Can do addition, subtraction, multiplication, and division without leaving the field GF(p) GF(2) = Mod 2 arithmetic GF(8) = Mod" xmpDM:startTime="2662233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="8 arithmetic There is no GF(6) since 6 is not a power of a prime." xmpDM:startTime="2662233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2662233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="GF(7) Multiplication Example  0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1" xmpDM:startTime="2776567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2" xmpDM:startTime="2776567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6" xmpDM:startTime="2776567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 6 5 4 3 2 1  0 1 2 3 4 5 6 0 0 0 0 0" xmpDM:startTime="2776567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3" xmpDM:startTime="2776567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1" xmpDM:startTime="2776567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="6 4 2 6 0 6 5 4 3 2 1" xmpDM:startTime="2776567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Polynomial Arithmetic f(x) = anxn + an-1xn-1 + … + a1x + a0 = ∑ aixi Ordinary polynomial arithmetic: Add," xmpDM:startTime="2796433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="subtract, multiply, divide polynomials, Find remainders, quotient." xmpDM:startTime="2796433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Some polynomials have no factors and are prime." xmpDM:startTime="2796433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Polynomial arithmetic with mod p coefficients Polynomial arithmetic with mod p coefficients and mod m(x) operations, where m(x) is a" xmpDM:startTime="2796433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="nth degree polynomial = GF(pn)" xmpDM:startTime="2796433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="GF(7) Multiplication Example  0 1 2 3 4 5 6 0 0 0 0 0 0 0 0 1" xmpDM:startTime="2805733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3 0 3 6 2" xmpDM:startTime="2805733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1 6 4 2 6" xmpDM:startTime="2805733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 6 5 4 3 2 1  0 1 2 3 4 5 6 0 0 0 0 0" xmpDM:startTime="2805733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 0 0 1 0 1 2 3 4 5 6 2 0 2 4 6 1 3 5 3" xmpDM:startTime="2805733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0 3 6 2 5 1 4 4 0 4 1 5 2 6 3 5 0 5 3 1" xmpDM:startTime="2805733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="6 4 2 6 0 6 5 4 3 2 1" xmpDM:startTime="2805733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Polynomial Arithmetic f(x) = anxn + an-1xn-1 + … + a1x + a0 = ∑ aixi Ordinary polynomial arithmetic: Add," xmpDM:startTime="2859533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="subtract, multiply, divide polynomials, Find remainders, quotient." xmpDM:startTime="2859533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Some polynomials have no factors and are prime." xmpDM:startTime="2859533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Polynomial arithmetic with mod p coefficients Polynomial arithmetic with mod p coefficients and mod m(x) operations, where m(x) is a" xmpDM:startTime="2859533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="nth degree polynomial = GF(pn)" xmpDM:startTime="2859533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Polynomial Arithmetic with Mod 2 Coefficients All coefficients are 0 or 1, e.g.," xmpDM:startTime="2992533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="let f(x) = x3 + x2 and g(x) = x2 + x + 1 f(x) + g(x) = x3 +" xmpDM:startTime="2992533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x + 1 f(x)  g(x) = x5 + x2 Polynomial Division: f(x) = q(x) g(x) + r(x) r(x) =" xmpDM:startTime="2992533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="remainder = f(x) mod g(x) if no remainder, say g(x) divides f(x) if g(x) has no divisors other than itself" xmpDM:startTime="2992533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="&amp; 1 say it is irreducible (or prime) polynomial Arithmetic modulo an irreducible polynomial forms a finite field Can use" xmpDM:startTime="2992533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euclid’s algorithm to find gcd and inverses." xmpDM:startTime="2992533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2992533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Example GF(23)" xmpDM:startTime="3227600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Computational Example in GF(2n) Since coefficients are 0 or 1, any polynomial can be represented as a bit string In" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="GF(23), (x2+1) is 1012 &amp; (x2+x+1) is 1112 Addition: (x2+1) + (x2+x+1) = x 101  111 = 0102 Multiplication:" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(x+1).(x2+1)" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x.(x2+1)" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ 1.(x2+1)" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x3+x+x2+1 = x3+x2+x+1 011.101" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 11112 Polynomial modulo reduction (get q(x) &amp; r(x)) is (x3+x2+x+1 ) mod (x3+x+1) = 1.(x3+x+1)" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ (x2) = x2 1111 mod 1011 = 1111  1011 = 01002" xmpDM:startTime="3644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4D Determine the gcd of the following pairs of polynomials over GF(11) 5x3+2x2-5x-2 and 5x5+2x4+6x2+9x" xmpDM:startTime="3743200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Computational Example in GF(2n) Since coefficients are 0 or 1, any polynomial can be represented as a bit string In" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="GF(23), (x2+1) is 1012 &amp; (x2+x+1) is 1112 Addition: (x2+1) + (x2+x+1) = x 101  111 = 0102 Multiplication:" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(x+1).(x2+1)" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x.(x2+1)" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ 1.(x2+1)" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x3+x+x2+1 = x3+x2+x+1 011.101" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 11112 Polynomial modulo reduction (get q(x) &amp; r(x)) is (x3+x2+x+1 ) mod (x3+x+1) = 1.(x3+x+1)" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ (x2) = x2 1111 mod 1011 = 1111  1011 = 01002" xmpDM:startTime="3823767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Example GF(23)" xmpDM:startTime="3824167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Computational Example in GF(2n) Since coefficients are 0 or 1, any polynomial can be represented as a bit string In" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="GF(23), (x2+1) is 1012 &amp; (x2+x+1) is 1112 Addition: (x2+1) + (x2+x+1) = x 101  111 = 0102 Multiplication:" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(x+1).(x2+1)" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x.(x2+1)" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ 1.(x2+1)" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x3+x+x2+1 = x3+x2+x+1 011.101" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 11112 Polynomial modulo reduction (get q(x) &amp; r(x)) is (x3+x2+x+1 ) mod (x3+x+1) = 1.(x3+x+1)" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="+ (x2) = x2 1111 mod 1011 = 1111  1011 = 01002" xmpDM:startTime="3866667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 4D Determine the gcd of the following pairs of polynomials over GF(11) 5x3+2x2-5x-2 and 5x5+2x4+6x2+9x" xmpDM:startTime="3867100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Using a Generator A generator g is an element whose powers generate all non-zero elements in F F={0, g0, g1," xmpDM:startTime="3867767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="…, gq-2} Can create generator from root of the irreducible polynomial then adding exponents of generator" xmpDM:startTime="3867767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Summary Euclid’s tabular method allows finding gcd and inverses Group is a set of element and an operation that satisfies" xmpDM:startTime="3938867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="closure, associativity, identity, and inverses Abelian group: Operation is commutative Rings have two operations: addition and multiplication Fields: Commutative rings" xmpDM:startTime="3938867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that have multiplicative identity and inverses Finite Fields or Galois Fields have pn elements where p is prime Polynomials with" xmpDM:startTime="3938867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="coefficients in GF(2n) also form a field." xmpDM:startTime="3938867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3938867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Lab 4: Brute Force Password Cracking Goal: Find user passwords from the password file." xmpDM:startTime="4076967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="This lab consists of using the following two tools: Password dump Pwdump7 to retrieve the password file, http://www.openwall.com/passwords/microsoft-windows-nt-2000-xp-2003-vista-7#pwdump" xmpDM:startTime="4076967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="John the ripper V1.8," xmpDM:startTime="4076967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Brute force password cracker to decode the entry http://www.openwall.com/john/" xmpDM:startTime="4076967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Throughout the lab, please note down the commands as indicated so that you can submit them as the solution." xmpDM:startTime="4076967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Remote desktop via VPN to CSE571XPS Use the common student account" xmpDM:startTime="4076967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Summary Euclid’s tabular method allows finding gcd and inverses Group is a set of element and an operation that satisfies" xmpDM:startTime="4079067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="closure, associativity, identity, and inverses Abelian group: Operation is commutative Rings have two operations: addition and multiplication Fields: Commutative rings" xmpDM:startTime="4079067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that have multiplicative identity and inverses Finite Fields or Galois Fields have pn elements where p is prime Polynomials with" xmpDM:startTime="4079067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="coefficients in GF(2n) also form a field." xmpDM:startTime="4079067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4079067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Lab 4: Brute Force Password Cracking Goal: Find user passwords from the password file." xmpDM:startTime="4202700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="This lab consists of using the following two tools: Password dump Pwdump7 to retrieve the password file, http://www.openwall.com/passwords/microsoft-windows-nt-2000-xp-2003-vista-7#pwdump" xmpDM:startTime="4202700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="John the ripper V1.8," xmpDM:startTime="4202700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Brute force password cracker to decode the entry http://www.openwall.com/john/" xmpDM:startTime="4202700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Throughout the lab, please note down the commands as indicated so that you can submit them as the solution." xmpDM:startTime="4202700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Remote desktop via VPN to CSE571XPS Use the common student account" xmpDM:startTime="4202700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Step 1: Get the Password File Read about pwdump Remote access the student account on CSE571XPS, open Command Prompt CD" xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="to c:/ Run pwdump7 -h to get some help Run pwdump7 with appropriate parameters to get the hash file from" xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="CSE571XPS." xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Note down the command you used." xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Open the hash file obtained in notepad." xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Delete all lines except the one with your last name." xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Save the file as c:\john180\run\&lt;your_last_name&gt;.txt" xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Delete the original full hash file that you downloaded" xmpDM:startTime="4446533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Step 2: Find Your Password CD to c:\john180\run Delete john.pot" xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="and john.log," xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if present." xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Run John to get help and read all the options Run John with the file you created in step 1" xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Your password is CseXXXX where X is a decimal digit [0-9]." xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Use correct options to search only for the specified pattern." xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Otherwise, John will take very long) If John takes more than one minute to finish then you have not chosen" xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="the correct options After John finishes." xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Note down the contents of john.pot" xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="file and submit." xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Delete your hash file, john.pot," xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="and john.log" xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4459800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3." xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Change Your Password Logout from the common student account and close your remote desktop connection Start a new remote desktop" xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="connection using your last name as username and the password you obtained in Step 2." xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Change your password to a stronger password of your choice." xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Do this from your own account (not the common student account)." xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Note the time and date you change the password." xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Submit the time as answer." xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Logout and close your remote desktop connection" xmpDM:startTime="4516600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Scan This to Download These Slides Raj Jain http://rajjain.com" xmpDM:startTime="4543367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4543367" xmpDM:duration="1000"/></rdf:li></rdf:Seq>
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                  <rdf:Description xmpDM:trackType="TableOfContents" xmpDM:frameRate="f1000" xmpDM:trackName="Table of Contents">
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                           <rdf:li><rdf:Description xmpDM:name="  1. Overview" xmpDM:startTime="17300" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="0" tscDM:imageoffset="0" tscDM:imagerect="0, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  2. Euclid's Algorithm" xmpDM:startTime="40733" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="1" tscDM:imageoffset="0" tscDM:imagerect="75, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  3. Euclid's Algorithm: Tabular Method" xmpDM:startTime="170033" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="2" tscDM:imageoffset="0" tscDM:imagerect="150, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  4. Euclid’s Algorithm Tabular Method (Cont)" xmpDM:startTime="573900" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="3" tscDM:imageoffset="0" tscDM:imagerect="225, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  5. Tabular Method (Cont)" xmpDM:startTime="619833" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="4" tscDM:imageoffset="0" tscDM:imagerect="300, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  6. Marker 1" xmpDM:startTime="897833" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="5" tscDM:imageoffset="0" tscDM:imagerect="375, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  7. Homework 4A" xmpDM:startTime="1038633" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="6" tscDM:imageoffset="0" tscDM:imagerect="450, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  8. Tabular Method (Cont)" xmpDM:startTime="1076400" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="7" tscDM:imageoffset="0" tscDM:imagerect="525, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="  9. Homework 4A" xmpDM:startTime="1216800" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="8" tscDM:imageoffset="0" tscDM:imagerect="600, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="10. Modular Arithmetic" xmpDM:startTime="1271467" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="9" tscDM:imageoffset="0" tscDM:imagerect="675, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="11. Modular Arithmetic Operations" xmpDM:startTime="1510500" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="10" tscDM:imageoffset="0" tscDM:imagerect="750, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="12. Modular Arithmetic" xmpDM:startTime="1561000" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="11" tscDM:imageoffset="0" tscDM:imagerect="825, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="13. Modular Arithmetic Operations" xmpDM:startTime="1573533" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="12" tscDM:imageoffset="0" tscDM:imagerect="900, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="14. Modular Arithmetic Properties" xmpDM:startTime="1740033" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="13" tscDM:imageoffset="0" tscDM:imagerect="975, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="15. Homework 4B" xmpDM:startTime="1904800" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="14" tscDM:imageoffset="0" tscDM:imagerect="1050, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="16. Group" xmpDM:startTime="2080767" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="15" tscDM:imageoffset="0" tscDM:imagerect="1125, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="17. Cyclic Group" xmpDM:startTime="2209967" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="16" tscDM:imageoffset="0" tscDM:imagerect="1200, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="18. Ring" xmpDM:startTime="2299267" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="17" tscDM:imageoffset="0" tscDM:imagerect="1275, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19. Homework 4C" xmpDM:startTime="2411500" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="18" tscDM:imageoffset="0" tscDM:imagerect="1350, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="20. Ring" xmpDM:startTime="2463100" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="19" tscDM:imageoffset="0" tscDM:imagerect="1425, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="21. Homework 4C" xmpDM:startTime="2473400" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="20" tscDM:imageoffset="0" tscDM:imagerect="1500, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="22. Field" xmpDM:startTime="2493200" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="21" tscDM:imageoffset="0" tscDM:imagerect="1575, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="23. Homework 4C" xmpDM:startTime="2494833" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="22" tscDM:imageoffset="0" tscDM:imagerect="1650, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="24. Ring" xmpDM:startTime="2495100" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="23" tscDM:imageoffset="0" tscDM:imagerect="1725, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="25. Homework 4C" xmpDM:startTime="2510700" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="24" tscDM:imageoffset="0" tscDM:imagerect="1800, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26. Field" xmpDM:startTime="2562933" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="25" tscDM:imageoffset="0" tscDM:imagerect="1875, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="27. Finite Fields or Galois Fields" xmpDM:startTime="2662233" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="26" tscDM:imageoffset="0" tscDM:imagerect="1950, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="28. GF(7) Multiplication Example " xmpDM:startTime="2776567" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="27" tscDM:imageoffset="0" tscDM:imagerect="2025, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="29. Polynomial Arithmetic" xmpDM:startTime="2796433" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="28" tscDM:imageoffset="0" tscDM:imagerect="2100, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="30. GF(7) Multiplication Example " xmpDM:startTime="2805733" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="29" tscDM:imageoffset="0" tscDM:imagerect="2175, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="31. Polynomial Arithmetic" xmpDM:startTime="2859533" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="30" tscDM:imageoffset="0" tscDM:imagerect="2250, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="32. Polynomial Arithmetic with Mod 2 Coefficients" xmpDM:startTime="2992533" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="31" tscDM:imageoffset="0" tscDM:imagerect="2325, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="33. Example GF(23)" xmpDM:startTime="3227600" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="32" tscDM:imageoffset="0" tscDM:imagerect="2400, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="34. Computational Example in GF(2n)" xmpDM:startTime="3644067" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="33" tscDM:imageoffset="0" tscDM:imagerect="2475, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="35. Homework 4D" xmpDM:startTime="3743200" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="34" tscDM:imageoffset="0" tscDM:imagerect="2550, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="36. Computational Example in GF(2n)" xmpDM:startTime="3823767" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="35" tscDM:imageoffset="0" tscDM:imagerect="2625, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="37. Example GF(23)" xmpDM:startTime="3824167" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="36" tscDM:imageoffset="0" tscDM:imagerect="2700, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="38. Computational Example in GF(2n)" xmpDM:startTime="3866667" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="37" tscDM:imageoffset="0" tscDM:imagerect="2775, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="39. Homework 4D" xmpDM:startTime="3867100" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="38" tscDM:imageoffset="0" tscDM:imagerect="2850, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="40. Using a Generator" xmpDM:startTime="3867767" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="39" tscDM:imageoffset="0" tscDM:imagerect="2925, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="41. Summary" xmpDM:startTime="3938867" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="40" tscDM:imageoffset="0" tscDM:imagerect="3000, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="42. Lab 4: Brute Force Password Cracking" xmpDM:startTime="4076967" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="41" tscDM:imageoffset="0" tscDM:imagerect="3075, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="43. Summary" xmpDM:startTime="4079067" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="42" tscDM:imageoffset="0" tscDM:imagerect="3150, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="44. Lab 4: Brute Force Password Cracking" xmpDM:startTime="4202700" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="43" tscDM:imageoffset="0" tscDM:imagerect="3225, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="45. Step 1: Get the Password File" xmpDM:startTime="4446533" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="44" tscDM:imageoffset="0" tscDM:imagerect="3300, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="46. Step 2: Find Your Password" xmpDM:startTime="4459800" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="45" tscDM:imageoffset="0" tscDM:imagerect="3375, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="47. 3. Change Your Password" xmpDM:startTime="4516600" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="46" tscDM:imageoffset="0" tscDM:imagerect="3450, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="48. Scan This to Download These Slides" xmpDM:startTime="4543367" tscDM:image="l_04nt_Thumbnails.png" tscDM:imageindex="47" tscDM:imageoffset="0" tscDM:imagerect="3525, 0, 75, 56"/></rdf:li></rdf:Seq>
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            <rdf:Description xmpDM:name="tscplayer">
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                  <rdf:Bag>
                     <rdf:li xmpDM:name="autohide" xmpDM:value="false"/><rdf:li xmpDM:name="autoplay" xmpDM:value="true"/><rdf:li xmpDM:name="loop" xmpDM:value="false"/><rdf:li xmpDM:name="searchable" xmpDM:value="true"/><rdf:li xmpDM:name="captionsenabled" xmpDM:value="false"/><rdf:li xmpDM:name="sidebarenabled" xmpDM:value="false"/><rdf:li xmpDM:name="unicodeenabled" xmpDM:value="false"/><rdf:li xmpDM:name="backgroundcolor" xmpDM:value="000000"/><rdf:li xmpDM:name="sidebarlocation" xmpDM:value="left"/><rdf:li xmpDM:name="endaction" xmpDM:value="jumptourl"/><rdf:li xmpDM:name="endactionparam" xmpDM:value="http://www.rajjain.com"/><rdf:li xmpDM:name="locale" xmpDM:value="en-US"/></rdf:Bag>
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               <tscDM:controllerText>
                  <rdf:Bag>
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         <tscDM:contentList>
            <rdf:Description>
               <tscDM:files>
                  <rdf:Seq>
                     <rdf:li xmpDM:name="0" xmpDM:value="l_04nt.mp4"/><rdf:li xmpDM:name="1" xmpDM:value="l_04nt_First_Frame.png"/><rdf:li xmpDM:name="2" xmpDM:value="l_04nt_Thumbnails.png"/></rdf:Seq>
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