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

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1.2 Efficient operations For the basic operations of +, −, ×, mod , div one may use standard “elementary school” algorithms reducing the work load by the following rules: a ⎧ ⎨ ⎩ + − × ⎫ ⎬ ⎭ b mod n = ⎛ ⎝(a mod n) ⎧ ⎨ ⎩ + − × ⎫ ⎬ ⎭ (b mod n) ⎞ ⎠ mod n The standard “elementary school” algorithms are precisely described in Knuth (Vol 2). For very large numbers, special purpose divide-and-conquer algorithms may be used for better efficiency of ×,mod, div. Consult the algorithmics book of Brassard-Bratley for these.
Page 1 and 2: COMP-547A /B Cryptography and Data
Page 3: Euclid Leonhard Euler Greatest Comm
Page 8 and 9: EXAMPLE 5 13 mod 7 = 5 8+4+1 mod 7
Page 10 and 11: Algorithm 1.2 ( Euclide gcd(a,b) )
Page 12 and 13: EXAMPLE gcd(7,19) x=1,y=0,x’=0,y
Page 14 and 15: 1.3 Solving linear congruentials A
Page 16 and 17: Similarily, when gcd(a, n) =g>1thes
Page 18 and 19: EXAMPLE 6x ≡ 5 (mod 9) has no sol
Page 20 and 21: EXAMPLE x ≡ 5 (mod 9) x ≡ 3 (mo
Page 22 and 23: 1.4.1 Legendre and Jacobi Symbols F
Page 24 and 25: Algorithm 1.3 ( Jacobi(a, n) ) 1:if
Page 26 and 27: EXAMPLE Jacobi(527, 723) = -Jacobi(
Page 28 and 29: 1.4.3 Extracting Square Roots modul
Page 30 and 31: EXAMPLE p = 37, a = 16 z = 3 (3 +
Page 36 and 37: 1.4.4 Extracting Square Roots modul
Page 38 and 39: EXAMPLE use CRT on the following 4
Page 40 and 41: EXAMPLE n = 37*43 = 1591, a = 477,
Page 42 and 43: Algorithm 1.5 ( Pseudo(a, n) ) 1: I
Page 44 and 45: EXAMPLE n = 1597, n-1 = 1596 = 399*
Page 46 and 47: EXAMPLE n = 1597, n-1 = 1596 = 399*
Page 48 and 49: EXAMPLE n = 1597 is either prime or
Page 50 and 51: 1.6 Quadratic Residuosity problem D
Page 52 and 53: Shafi Goldwasser Silvio Micali This
Page 54 and 55:
2 Finite Fields 2.1 Prime Fields Le
Page 56 and 57:
2.1.1 Primitive Elements In all fin
Page 58 and 59:
Relation to Quadratic residues As a
Page 60 and 61:
Algorithm 2.2 ( Kalai randfact(n) )
Page 62 and 63:
EXAMPLE n = 10000000 S = [1710521,
Page 64 and 65:
2.2 Polynomials over a field Apolyn
Page 66 and 67:
2.2.1 Irreducible Polynomials Apoly
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Algorithm 2.3 ( Rabin Irr(p, n) ) 1
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Examples computations over F 2 5 10
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COMP-547A /B Cryptography and Data
Page 74 and 75:
1.4.5 ∗∗∗ Extracting Square R
Page 76 and 77:
The following value of k will achei
Page 78 and 79:
2.4 Application of finite fields: S
Page 80 and 81:
Let’s be a bit more formal. Let S
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Course notes (chap. 1 Number Theory, chap. 2 ... - McGill University

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