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

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EXAMPLE 5 13 mod 7 = 5 8+4+1 mod 7 = 5 8 x5 4 x5 1 mod 7 5 0 , 5 1 , 5 2 , 5 4 , 5 8 ≡ 1, 5, 4, 2, 4 (mod 7) = 4x2x5 mod 7 = 1x5 mod 7 = 5 
1.2.2 GCD calculations and multiplicative inverses Note. gcd(a, b) =g →∃ x,y ∈ Z such that g = ax + by. The following recursive definition is based on the property gcd(a, b) =gcd(a, b − a). gcd(a, b) = { a if b =0 gcd(b, a mod b) otherwise The idea behind the following iterative algorithm is to maintain in each iteration the relations g = ax+by and g ′ = ax ′ +by ′ while reducing the value of g. At the end of the algorithm, the value of g is gcd(a, b). The final value of x is such that ax ≡ g (mod b) andbysymmetry,thefinalvalueofy is such that by ≡ g (mod a) . Whengcd(a, b) =1,wefindthatx is the multiplicative inverse of a modulo b and that y is the multiplicative inverse of b modulo a.
Page 1 and 2: COMP-547A /B Cryptography and Data
Page 3 and 4: Euclid Leonhard Euler Greatest Comm
Page 7: 1.2.1 Fast modular exponentiation T
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 and 20: 1.3.1 Chinese Remainder Theorem Qin
Page 21 and 22: 1.4 Quadratic Residues Quadratic re
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+
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EXTRA SLIDES
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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
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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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