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                           <rdf:li><rdf:Description xmpDM:name="Number Theory Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse571-17/" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Overview Prime numbers Fermat’s and Euler’s Theorems Testing for primality The Chinese Remainder Theorm Discrete Logarithms These slides are partly" xmpDM:startTime="3167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="based on Lawrie Brown’s slides supplied with William Stallings’s book “Cryptography and Network Security: Principles and Practice,” 7th Ed, 2017." xmpDM:startTime="3167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Fermat's Little Theorem Given a prime number p: ap-1 = 1 (mod p) For all integers ap Or ap =" xmpDM:startTime="30500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="a (mod p) Example: 14 mod 5=1 24 mod 5=1 34 mod 5=1 44 mod 5=1" xmpDM:startTime="30500" 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="127900" 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="127900" 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="127900" 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="127900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for p.q" xmpDM:startTime="127900" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(p,q prime) ø(p.q)=(p-1)x(q-1)" xmpDM:startTime="127900" 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="127900" 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="513100" 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="513100" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="11 Also have: aø(n)+1 = a (mod n)" xmpDM:startTime="513100" 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="528933" 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="528933" 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="528933" 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="528933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for p.q" xmpDM:startTime="528933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(p,q prime) ø(p.q)=(p-1)x(q-1)" xmpDM:startTime="528933" 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="528933" 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="536200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="n=10; 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0.99999" xmpDM:startTime="806533" 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="806533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="806533" 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="810500" 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="810500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="11 Also have: aø(n)+1 = a (mod n)" xmpDM:startTime="810500" 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="811400" 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="811400" 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="811400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if aq mod n = 1 then return (“inconclusive&quot;); 4." xmpDM:startTime="811400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for j = 0 to k – 1 do 5." xmpDM:startTime="811400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="if (a2jq mod n = n-1) then return(“inconclusive&quot;) 6." xmpDM:startTime="811400" 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="811400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E.g.," xmpDM:startTime="811400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="for t=10 this probability is &gt; 0.99999" xmpDM:startTime="811400" 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="811400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Y mod n = n-1? aq mod n = n-1? Y" xmpDM:startTime="811400" 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="1380667" 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="1380667" 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="1380667" 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="1433767" 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="1433767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="0.5" xmpDM:startTime="1433767" 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="1433767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Other times are quite far apart" xmpDM:startTime="1433767" 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="1573833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod M = m1m2..mk" xmpDM:startTime="1573833" 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="1573833" 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="1573833" 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="1573833" 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="1573833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod 7 Þ x=1" xmpDM:startTime="1573833" xmpDM:duration="1000"/></rdf:li></rdf:Seq>
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