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

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2.2.1 Irreducible Polynomials Apolynomialg(x) isirreducible (Maple irreduc) ifitisnottheproductof two polynomials h(x),k(x) oflowerdegrees. Weusethefollowingtheorem to find irreducible polynomials. Theorem 2.6 Let l 1 ,l 2 ,...,l k be the prime factors of n and m i = n/l i for 1 ≤ i ≤ k. Apolynomialg(x) of degree n is irreducible over F p iff • g(x)|x pn − x • gcd(g(x),x pm i − x) =1for 1 ≤ i ≤ k
F 2 x +1 x 9 + x 4 +1 x 2 + x +1 x 10 + x 3 +1 x 3 + x +1 x 11 + x 2 +1 x 4 + x +1 x 12 + x 6 + x 4 + x +1 x 5 + x 2 +1 x 13 + x 4 + x 3 + x +1 x 6 + x +1 x 14 + x 10 + x 6 + x +1 x 7 + x 3 +1 x 15 + x +1 x 8 + x 4 + x 3 + x 2 +1 x 16 + x 12 + x 3 + x +1 Figure 1: Irreducible polynomials over F 2 . F 3 F 5 F 7 x +1 x +1 x +1 x 2 + x +2 x 2 + x +2 x 2 + x +3 x 3 +2x +1 x 3 +3x +2 x 3 +3x +2 x 4 + x +2 x 4 + x 2 + x +2 x 5 +2x +1 x 6 + x +2 Figure 2: Irreducible polynomials over F 3 ,F 5 ,F 7 .
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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
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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: We also have operations g(x) modh(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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