Course notes (chap. 1 Number Theory, chap. 2 ... - McGill University

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EXAMPLE x ≡ 5 (mod 9) x ≡ 3 (mod 4)   x ≡ 7 (mod 13) has a unique solution mod 468  = 4∙9∙13 x = 5∙52∙(52 -1 mod 9)+ 3∙117∙(117 -1 mod 4)    + 7∙36∙(36 -1 mod 13) x = 260∙4 + 351∙1 + 252∙4 x = 1040 + 351 + 1008 x = 2399 x ≡ 59 (mod 468)
1.4 Quadratic Residues Quadratic residues modulo n are the integers with an integer square root modulo n (Maple quadres): QR n = {a :gcd(a, n) =1, ∃r[a ≡ r 2 QNR n = {a :gcd(a, n) =1, ∀r[a ≢ r 2 (mod n)]} (mod n)]} Example: since QR 17 = {1, 2, 4, 8, 9, 13, 15, 16} QNR 17 = {3, 5, 6, 7, 10, 11, 12, 14} {1 2 , 2 2 , 3 2 , 4 2 , 5 2 , 6 2 , 7 2 , 8 2 , 9 2 , 10 2 , 11 2 , 12 2 , 13 2 , 14 2 , 15 2 , 16 2 }≡ {1, 2, 4, 8, 9, 13, 15, 16} (mod 17). Theorem 1.2 Let p be an odd prime number #QR p =#QNR p =(p − 1)/2.
Page 1 and 2: COMP-547A /B Cryptography and Data
Page 3 and 4: Euclid Leonhard Euler Greatest Comm
Page 7 and 8: 1.2.1 Fast modular exponentiation T
Page 9 and 10: 1.2.2 GCD calculations and multipli
Page 11 and 12: EXAMPLE gcd(7,19) = gcd(19,7) = gcd
Page 13 and 14: EXAMPLE gcd(7,19) = gcd(1,0) = 1 x
Page 15 and 16: EXAMPLE 7x ≡ 5 (mod 9) 7 -1 ∙7x
Page 17 and 18: Finally, we know that a solution x
Page 19: 1.3.1 Chinese Remainder Theorem Qin
Page 23 and 24: Properties ( 1 =+1 n) ( ) ab ( a )
Page 25 and 26: EXAMPLE Jacobi(527, 723) = -Jacobi(
Page 27 and 28: Pierre de Fermat 1.4.2 Fermat-Euler
Page 29 and 30: For prime numbers p ≡ 1(mod4)anda
Page 31 and 32: EXAMPLE p = 37, a = 16 z = 5 (5 +
Page 37 and 38: EXAMPLE n = 37*43 = 1591, a = 16
Page 39 and 40: Definition 1.7 (SQROOT) The square
Page 41 and 42: 1.5 Prime numbers If we want a rand
Page 43 and 44: EXAMPLE n = 37*43 = 1591, n-1 = 159
Page 45 and 46: EXAMPLE n = 1597, n-1 = 1596 = 399*
Page 47 and 48: Algorithm 1.6 ( Miller-Rabin prime(
Page 49 and 50: In August of 2002, Agrawal, Kayal,
Page 51 and 52: Remark: If n is a prime, then it is
Page 53 and 54: 3: Pick x ∈ R Z n ∗ 4: c i ←
Page 55 and 56: Examples F 2 =({0, 1}, ⊕, ∧). F
Page 57 and 58: Algorithm 2.1 ( Primitive(q) ) 1: L
Page 59 and 60: Factoring q − 1... The only effic
Page 61 and 62: EXAMPLE n = 10000000 S = [2710128,
Page 63 and 64: EXAMPLE n = 10000000 S = [4721693,
Page 65 and 66: We also have operations g(x) modh(x
Page 67 and 68: F 2 x +1 x 9 + x 4 +1 x 2 + x +1 x
Page 69 and 70: 2.3 General Fields Let p be a prime
Page 71 and 72:
10011×01110 = 01001 since (x 4 +x+
Page 73 and 74:
EXTRA SLIDES
Page 75 and 76:
First, consider the case e =2. Sinc
Page 77 and 78:
Second, notice that the same exact
Page 79 and 80:
Suppose Alice wants to distribute a
Page 81:
Algorithm 2.6 ( Solve(x 1 ,x 2 ,...
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Course notes (chap. 1 Number Theory, chap. 2 ... - McGill University

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