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

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1.4.3 Extracting Square Roots modulo p Theorem 1.6 For prime numbers p ≡ 3(mod4)and a ∈ QR p ,wehave that r = a (p+1)/4 mod p is a square root of a. Proof. (a (p+1)/4) ) 2 ≡ a (p−1)/2 · a (mod p) ≡ a (mod p)(Fermat, sec. 1.3)
For prime numbers p ≡ 1(mod4)anda ∈ QR p ,there(only)existsan efficient probabilistic algorithm. We present one found in the algorithmics book of Brassard-Bratley: Algorithm 1.4 ( rootLV(a, p, VAR r, VAR success) ) 1: z ← uniform(1 ...p− 1) 2: IF a = z 2 mod p {very unlikely} THEN success ← true, r ← z 3: ELSE compute c and d such that 0 ≤ c ≤ p − 1, 0 ≤ d ≤ p − 1, and c + d √ a ≡ (z + √ a) (p−1)/2 mod p 4: IF d =0THEN success ← false 5: ELSE (c =0), success ← true, Michele Cipolla 6: compute r such that 1 ≤ r ≤ p − 1andd · r ≡ 1modp
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 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: Pierre de Fermat 1.4.2 Fermat-Euler
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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