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                           <rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="173267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="173267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="173267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Prime Distribution Prime numbers: 1 2 3 5 7 11 13 17 19 23 29 31 Prime number theorem states" xmpDM:startTime="180533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that primes occur roughly every (ln n) integers But can immediately ignore even numbers So in practice need only test" xmpDM:startTime="180533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0.5" xmpDM:startTime="180533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ln(n) numbers of size n to locate a prime Note this is only the “average” Sometimes primes are close together" xmpDM:startTime="180533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Other times are quite far apart" xmpDM:startTime="180533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="184367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="184367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="184367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="239467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="253467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="253467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="253467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Prime Distribution Prime numbers: 1 2 3 5 7 11 13 17 19 23 29 31 Prime number theorem states" xmpDM:startTime="279800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that primes occur roughly every (ln n) integers But can immediately ignore even numbers So in practice need only test" xmpDM:startTime="279800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0.5" xmpDM:startTime="279800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ln(n) numbers of size n to locate a prime Note this is only the “average” Sometimes primes are close together" xmpDM:startTime="279800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Other times are quite far apart" xmpDM:startTime="279800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="280933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="280933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="280933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="281200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="291967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="291967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="291967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="320367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="332467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="332467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="332467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="350133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="373100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="373100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="373100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="414400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="439200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="439200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="439200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Prime Distribution Prime numbers: 1 2 3 5 7 11 13 17 19 23 29 31 Prime number theorem states" xmpDM:startTime="445800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that primes occur roughly every (ln n) integers But can immediately ignore even numbers So in practice need only test" xmpDM:startTime="445800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0.5" xmpDM:startTime="445800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ln(n) numbers of size n to locate a prime Note this is only the “average” Sometimes primes are close together" xmpDM:startTime="445800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Other times are quite far apart" xmpDM:startTime="445800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem If working modulo a product of numbers E.g.," xmpDM:startTime="528700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod M = m1m2..mk" xmpDM:startTime="528700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder theorem lets us work in each moduli mi separately Since computational cost is proportional to size, this is" xmpDM:startTime="528700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="faster Example: 452 mod 105 = (452 mod 3)(105/3){(105/3)-1 mod 3} +(452 mod 5)(105/5){(105/5)-1 mod 5} +(452 mod 7)(105/7){(105/7)-1 mod" xmpDM:startTime="528700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="7} = 235(35-1 mod 3) +2x21(21-1 mod 5) +415(15-1 mod 7) = 2352 +2211 +4151 = (140+42+60) mod 105 =" xmpDM:startTime="528700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="242 mod 105 = 32 35-1 =x mod 3 35x=1 mod 3 Þ x=2 21x=1 mod 5 Þ x=1 15x=1" xmpDM:startTime="528700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod 7 Þ x=1" xmpDM:startTime="528700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Alternately, the solution to the following equations: x = a1 mod m1 x = a2 mod m2" xmpDM:startTime="1019633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = ak mod mk where m1, m2, ..." xmpDM:startTime="1019633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name=", mk are relatively prime is found as follows: M = m1 m2 ..." xmpDM:startTime="1019633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mk then" xmpDM:startTime="1019633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Example For a parade, marchers are arranged in columns of seven, but one person is left out." xmpDM:startTime="1107800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="In columns of eight, two people are left out." xmpDM:startTime="1107800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="With columns of nine, three people are left out." xmpDM:startTime="1107800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="How many marchers are there? Ref: http://demonstrations.wolfram.com/ChineseRemainderTheorem/" xmpDM:startTime="1107800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1107800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Alternately, the solution to the following equations: x = a1 mod m1 x = a2 mod m2" xmpDM:startTime="1191400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = ak mod mk where m1, m2, ..." xmpDM:startTime="1191400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name=", mk are relatively prime is found as follows: M = m1 m2 ..." xmpDM:startTime="1191400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mk then" xmpDM:startTime="1191400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Example For a parade, marchers are arranged in columns of seven, but one person is left out." xmpDM:startTime="1193300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="In columns of eight, two people are left out." xmpDM:startTime="1193300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="With columns of nine, three people are left out." xmpDM:startTime="1193300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="How many marchers are there? Ref: http://demonstrations.wolfram.com/ChineseRemainderTheorem/" xmpDM:startTime="1193300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1193300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Primitive Roots From Euler’s theorem have aø(n)mod n=1 Consider am = 1 (mod n), GCD(a,n)=1 For some a’s, m can" xmpDM:startTime="1308333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="smaller than ø(n) If the smallest m is ø(n) then a is called a primitive root If n is prime," xmpDM:startTime="1308333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="then successive powers of a &quot;generate&quot; the group mod n These are useful but relatively hard to find" xmpDM:startTime="1308333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Example For a parade, marchers are arranged in columns of seven, but one person is left out." xmpDM:startTime="1393933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="In columns of eight, two people are left out." xmpDM:startTime="1393933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="With columns of nine, three people are left out." xmpDM:startTime="1393933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="How many marchers are there? Ref: http://demonstrations.wolfram.com/ChineseRemainderTheorem/" xmpDM:startTime="1393933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1393933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Alternately, the solution to the following equations: x = a1 mod m1 x = a2 mod m2" xmpDM:startTime="1394600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = ak mod mk where m1, m2, ..." xmpDM:startTime="1394600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name=", mk are relatively prime is found as follows: M = m1 m2 ..." xmpDM:startTime="1394600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mk then" xmpDM:startTime="1394600" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem If working modulo a product of numbers E.g.," xmpDM:startTime="1395133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod M = m1m2..mk" xmpDM:startTime="1395133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder theorem lets us work in each moduli mi separately Since computational cost is proportional to size, this is" xmpDM:startTime="1395133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="faster Example: 452 mod 105 = (452 mod 3)(105/3){(105/3)-1 mod 3} +(452 mod 5)(105/5){(105/5)-1 mod 5} +(452 mod 7)(105/7){(105/7)-1 mod" xmpDM:startTime="1395133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="7} = 235(35-1 mod 3) +2x21(21-1 mod 5) +415(15-1 mod 7) = 2352 +2211 +4151 = (140+42+60) mod 105 =" xmpDM:startTime="1395133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="242 mod 105 = 32 35-1 =x mod 3 35x=1 mod 3 Þ x=2 21x=1 mod 5 Þ x=1 15x=1" xmpDM:startTime="1395133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod 7 Þ x=1" xmpDM:startTime="1395133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Prime Distribution Prime numbers: 1 2 3 5 7 11 13 17 19 23 29 31 Prime number theorem states" xmpDM:startTime="1395567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that primes occur roughly every (ln n) integers But can immediately ignore even numbers So in practice need only test" xmpDM:startTime="1395567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0.5" xmpDM:startTime="1395567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ln(n) numbers of size n to locate a prime Note this is only the “average” Sometimes primes are close together" xmpDM:startTime="1395567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Other times are quite far apart" xmpDM:startTime="1395567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="1396100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="1396100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="1396100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="1396633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euler's Theorem A generalisation of Fermat's Theorem aø(n) = 1 (mod n) for any a, n where gcd(a,n)=1 Example: a=3;" xmpDM:startTime="1397200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="n=10; ø(10)=4; hence 34 = 81 = 1 mod 10 a=2; n=11; ø(11)=10; hence 210 = 1024 = 1 mod" xmpDM:startTime="1397200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="11 Also have: aø(n)+1 = a (mod n)" xmpDM:startTime="1397200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="1421700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="1422000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="1422000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="1422000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Prime Distribution Prime numbers: 1 2 3 5 7 11 13 17 19 23 29 31 Prime number theorem states" xmpDM:startTime="1422667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that primes occur roughly every (ln n) integers But can immediately ignore even numbers So in practice need only test" xmpDM:startTime="1422667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0.5" xmpDM:startTime="1422667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ln(n) numbers of size n to locate a prime Note this is only the “average” Sometimes primes are close together" xmpDM:startTime="1422667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Other times are quite far apart" xmpDM:startTime="1422667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem If working modulo a product of numbers E.g.," xmpDM:startTime="1423500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod M = m1m2..mk" xmpDM:startTime="1423500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder theorem lets us work in each moduli mi separately Since computational cost is proportional to size, this is" xmpDM:startTime="1423500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="faster Example: 452 mod 105 = (452 mod 3)(105/3){(105/3)-1 mod 3} +(452 mod 5)(105/5){(105/5)-1 mod 5} +(452 mod 7)(105/7){(105/7)-1 mod" xmpDM:startTime="1423500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="7} = 235(35-1 mod 3) +2x21(21-1 mod 5) +415(15-1 mod 7) = 2352 +2211 +4151 = (140+42+60) mod 105 =" xmpDM:startTime="1423500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="242 mod 105 = 32 35-1 =x mod 3 35x=1 mod 3 Þ x=2 21x=1 mod 5 Þ x=1 15x=1" xmpDM:startTime="1423500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod 7 Þ x=1" xmpDM:startTime="1423500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Alternately, the solution to the following equations: x = a1 mod m1 x = a2 mod m2" xmpDM:startTime="1424200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = ak mod mk where m1, m2, ..." xmpDM:startTime="1424200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name=", mk are relatively prime is found as follows: M = m1 m2 ..." xmpDM:startTime="1424200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mk then" xmpDM:startTime="1424200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Example For a parade, marchers are arranged in columns of seven, but one person is left out." xmpDM:startTime="1424733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="In columns of eight, two people are left out." xmpDM:startTime="1424733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="With columns of nine, three people are left out." xmpDM:startTime="1424733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="How many marchers are there? Ref: http://demonstrations.wolfram.com/ChineseRemainderTheorem/" xmpDM:startTime="1424733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1424733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Primitive Roots From Euler’s theorem have aø(n)mod n=1 Consider am = 1 (mod n), GCD(a,n)=1 For some a’s, m can" xmpDM:startTime="1425700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="smaller than ø(n) If the smallest m is ø(n) then a is called a primitive root If n is prime," xmpDM:startTime="1425700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="then successive powers of a &quot;generate&quot; the group mod n These are useful but relatively hard to find" xmpDM:startTime="1425700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Powers mod 19 2, 3, 10, 13, 14, 15 are primitive roots of 19" xmpDM:startTime="1437467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is" xmpDM:startTime="1580733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="to find i such that b = ai (mod p) This is written as i = dloga b (mod p)" xmpDM:startTime="1580733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If a is a primitive root then it always exists, otherwise it may not, e.g.," xmpDM:startTime="1580733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive" xmpDM:startTime="1580733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="powers While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem" xmpDM:startTime="1580733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Powers mod 19 2, 3, 10, 13, 14, 15 are primitive roots of 19" xmpDM:startTime="1652300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is" xmpDM:startTime="1719400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="to find i such that b = ai (mod p) This is written as i = dloga b (mod p)" xmpDM:startTime="1719400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If a is a primitive root then it always exists, otherwise it may not, e.g.," xmpDM:startTime="1719400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive" xmpDM:startTime="1719400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="powers While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem" xmpDM:startTime="1719400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Powers mod 19 2, 3, 10, 13, 14, 15 are primitive roots of 19" xmpDM:startTime="1770467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is" xmpDM:startTime="1797433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="to find i such that b = ai (mod p) This is written as i = dloga b (mod p)" xmpDM:startTime="1797433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If a is a primitive root then it always exists, otherwise it may not, e.g.," xmpDM:startTime="1797433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive" xmpDM:startTime="1797433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="powers While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem" xmpDM:startTime="1797433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms mod 19" xmpDM:startTime="1892467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is" xmpDM:startTime="1916733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="to find i such that b = ai (mod p) This is written as i = dloga b (mod p)" xmpDM:startTime="1916733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If a is a primitive root then it always exists, otherwise it may not, e.g.," xmpDM:startTime="1916733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive" xmpDM:startTime="1916733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="powers While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem" xmpDM:startTime="1916733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Powers mod 19 2, 3, 10, 13, 14, 15 are primitive roots of 19" xmpDM:startTime="1917200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is" xmpDM:startTime="1929267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="to find i such that b = ai (mod p) This is written as i = dloga b (mod p)" xmpDM:startTime="1929267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If a is a primitive root then it always exists, otherwise it may not, e.g.," xmpDM:startTime="1929267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive" xmpDM:startTime="1929267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="powers While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem" xmpDM:startTime="1929267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms mod 19" xmpDM:startTime="1942867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1943433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms mod 19" xmpDM:startTime="2068300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Discrete Logarithms The inverse problem to exponentiation is to find the discrete logarithm of a number modulo p That is" xmpDM:startTime="2068533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="to find i such that b = ai (mod p) This is written as i = dloga b (mod p)" xmpDM:startTime="2068533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If a is a primitive root then it always exists, otherwise it may not, e.g.," xmpDM:startTime="2068533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = log3 4 mod 13 has no answer x = log2 3 mod 13 = 4 by trying successive" xmpDM:startTime="2068533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="powers While exponentiation is relatively easy, finding discrete logarithms is generally a hard problem" xmpDM:startTime="2068533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Powers mod 19 2, 3, 10, 13, 14, 15 are primitive roots of 19" xmpDM:startTime="2068667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Primitive Roots From Euler’s theorem have aø(n)mod n=1 Consider am = 1 (mod n), GCD(a,n)=1 For some a’s, m can" xmpDM:startTime="2068800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="smaller than ø(n) If the smallest m is ø(n) then a is called a primitive root If n is prime," xmpDM:startTime="2068800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="then successive powers of a &quot;generate&quot; the group mod n These are useful but relatively hard to find" xmpDM:startTime="2068800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Example For a parade, marchers are arranged in columns of seven, but one person is left out." xmpDM:startTime="2068933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="In columns of eight, two people are left out." xmpDM:startTime="2068933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="With columns of nine, three people are left out." xmpDM:startTime="2068933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="How many marchers are there? Ref: http://demonstrations.wolfram.com/ChineseRemainderTheorem/" xmpDM:startTime="2068933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2068933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem Alternately, the solution to the following equations: x = a1 mod m1 x = a2 mod m2" xmpDM:startTime="2069100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="x = ak mod mk where m1, m2, ..." xmpDM:startTime="2069100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name=", mk are relatively prime is found as follows: M = m1 m2 ..." xmpDM:startTime="2069100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mk then" xmpDM:startTime="2069100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder Theorem If working modulo a product of numbers E.g.," xmpDM:startTime="2069333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod M = m1m2..mk" xmpDM:startTime="2069333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Chinese Remainder theorem lets us work in each moduli mi separately Since computational cost is proportional to size, this is" xmpDM:startTime="2069333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="faster Example: 452 mod 105 = (452 mod 3)(105/3){(105/3)-1 mod 3} +(452 mod 5)(105/5){(105/5)-1 mod 5} +(452 mod 7)(105/7){(105/7)-1 mod" xmpDM:startTime="2069333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="7} = 235(35-1 mod 3) +2x21(21-1 mod 5) +415(15-1 mod 7) = 2352 +2211 +4151 = (140+42+60) mod 105 =" xmpDM:startTime="2069333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="242 mod 105 = 32 35-1 =x mod 3 35x=1 mod 3 Þ x=2 21x=1 mod 5 Þ x=1 15x=1" xmpDM:startTime="2069333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod 7 Þ x=1" xmpDM:startTime="2069333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Prime Distribution Prime numbers: 1 2 3 5 7 11 13 17 19 23 29 31 Prime number theorem states" xmpDM:startTime="2069467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="that primes occur roughly every (ln n) integers But can immediately ignore even numbers So in practice need only test" xmpDM:startTime="2069467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0.5" xmpDM:startTime="2069467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ln(n) numbers of size n to locate a prime Note this is only the “average” Sometimes primes are close together" xmpDM:startTime="2069467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Other times are quite far apart" xmpDM:startTime="2069467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example Test 29 for primality 29-1 = 28 = 227 = 2kq Þ k=2, q=7 Let a" xmpDM:startTime="2069667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= 10 107 mod 29 = 17 102×7 mod 29 =172 mod 29 = 28 Þ Inconclusive Test 221 for" xmpDM:startTime="2069667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="primality 221-1=220=22 55 Let a=5 555 mod 221 =112 52×55 mod 221 = 1122 mod 221 =168 Þ Composite" xmpDM:startTime="2069667" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality A test for large primes based on Fermat’s Theorem TEST (n) is: 1." xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find integers k, q, k &gt; 0, q odd, so that (n–1)=2kq 2." xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Select a random integer a, 1&lt;a&lt;n–1 3." xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="return (“composite&quot;) If inconclusive after t tests with different a’s: Probability (n is Prime after t tests) = 1- 4-t" xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="aq mod n = 1? (aq)2 mod n = n-1? (aq)4 mod n = n-1? Inconclusive Composite Y Y Y" xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="2070100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euler's Theorem A generalisation of Fermat's Theorem aø(n) = 1 (mod n) for any a, n where gcd(a,n)=1 Example: a=3;" xmpDM:startTime="2070633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="n=10; ø(10)=4; hence 34 = 81 = 1 mod 10 a=2; n=11; ø(11)=10; hence 210 = 1024 = 1 mod" xmpDM:startTime="2070633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="11 Also have: aø(n)+1 = a (mod n)" xmpDM:startTime="2070633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euler Totient Function ø(n) When doing arithmetic modulo n complete set of residues is: 0..n-1" xmpDM:startTime="2071167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Reduced set of residues is those residues which are relatively prime to n, e.g.," xmpDM:startTime="2071167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for n=10, complete set of residues is {0,1,2,3,4,5,6,7,8,9} reduced set of residues is {1,3,7,9} Number of elements in reduced set" xmpDM:startTime="2071167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="of residues is called the Euler Totient Function ø(n) In general need prime factorization, but for p (p prime) ø(p)=p-1" xmpDM:startTime="2071167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for p.q" xmpDM:startTime="2071167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(p,q prime) ø(p.q)=(p-1)x(q-1)" xmpDM:startTime="2071167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Examples: ø(37) = 36 ø(21) = (3–1)x(7–1) = 2x6 = 12" xmpDM:startTime="2071167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Euler Totient Function ø(n) When doing arithmetic modulo n complete set of residues is: 0..n-1" xmpDM:startTime="2157900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Reduced set of residues is those residues which are relatively prime to n, e.g.," xmpDM:startTime="2157900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for n=10, complete set of residues is {0,1,2,3,4,5,6,7,8,9} reduced set of residues is {1,3,7,9} Number of elements in reduced set" xmpDM:startTime="2157900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="of residues is called the Euler Totient Function ø(n) In general need prime factorization, but for p (p prime) ø(p)=p-1" xmpDM:startTime="2157900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for p.q" xmpDM:startTime="2157900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(p,q prime) ø(p.q)=(p-1)x(q-1)" xmpDM:startTime="2157900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Examples: ø(37) = 36 ø(21) = (3–1)x(7–1) = 2x6 = 12" xmpDM:startTime="2157900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Acronyms AD Anno Domini (Latin for &quot;The Year of the Lord&quot;) CRT Chinese Remainder Theorem DSA Digital signature algorithm GCD" xmpDM:startTime="2158500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Greatest Common Divisor RSA Rivest, Samir, and Adleman" xmpDM:startTime="2158500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Scan This to Download These Slides Raj Jain http://rajjain.com" xmpDM:startTime="2159067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2159067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Related Modules Audio/Video Recordings and Podcasts of Professor Raj Jain's Lectures, https://www.youtube.com/channel/UCN4-5wzNP9-ruOzQMs-8NUw" xmpDM:startTime="2159567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="CSE473S: Introduction to Computer Networks (Fall 2016), http://www.cse.wustl.edu/~jain/cse473-16/index.html" xmpDM:startTime="2159567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Wireless and Mobile Networking (Spring 2016), http://www.cse.wustl.edu/~jain/cse574-16/index.html" xmpDM:startTime="2159567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="CSE571S: Network Security (Fall 2014), http://www.cse.wustl.edu/~jain/cse571-14/index.html" xmpDM:startTime="2159567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="CSE571S: Network Security (Spring 2017), http://www.cse.wustl.edu/~jain/cse571-17/index.html" xmpDM:startTime="2159567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2159567" xmpDM:duration="1000"/></rdf:li></rdf:Seq>
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                           <rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example" xmpDM:startTime="173267" tscDM:image="l_03mnt2_Thumbnails.png" tscDM:imageindex="0" tscDM:imageoffset="0" tscDM:imagerect="0, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Prime Distribution" xmpDM:startTime="180533" tscDM:image="l_03mnt2_Thumbnails.png" tscDM:imageindex="1" tscDM:imageoffset="0" tscDM:imagerect="75, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example" xmpDM:startTime="184367" tscDM:image="l_03mnt2_Thumbnails.png" tscDM:imageindex="2" tscDM:imageoffset="0" tscDM:imagerect="150, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm for Primality" xmpDM:startTime="239467" tscDM:image="l_03mnt2_Thumbnails.png" tscDM:imageindex="3" tscDM:imageoffset="0" tscDM:imagerect="225, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Miller Rabin Algorithm Example" xmpDM:startTime="253467" 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