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

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1 Basic <strong>Number</strong> <strong>Theory</strong> 1.1 Definitions Divisibility: a|b ⇐⇒ ∃k ∈ Z [b = ak] Congruences: a ≡ b (mod n) ⇐⇒ n|(a − b) Modulo operator: (Maple irem, mod) b mod n =min{a ≥ 0:a ≡ b (mod n)} Division operator: (Maple iquo) b div n = ⌊b/n⌋ = b − (b mod n) n a ≡ b (mod n) ⟺ a mod n = b mod n
Euclid Leonhard Euler Greatest Common Divider: (Maple igcd, igcdex) g =gcd(a, b) ⇐⇒ g|a, g|b and [g ′ |a, g ′ |b ⇒ g ′ |g] Euler’s Phi function: (Maple phi) φ(n) =#{a :0
Page 1: COMP-547A /B Cryptography and Data
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 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 ←
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Examples F 2 =({0, 1}, ⊕, ∧). F
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Algorithm 2.1 ( Primitive(q) ) 1: L
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Factoring q − 1... The only effic
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EXAMPLE n = 10000000 S = [2710128,
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EXAMPLE n = 10000000 S = [4721693,
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We also have operations g(x) modh(x
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F 2 x +1 x 9 + x 4 +1 x 2 + x +1 x
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2.3 General Fields Let p be a prime
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10011×01110 = 01001 since (x 4 +x+
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EXTRA SLIDES
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First, consider the case e =2. Sinc
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Second, notice that the same exact
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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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