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

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2.4 Application of finite fields: Secret Sharing ApolynomialoverF q is specified by a finite sequence (a n ,a n−1 ,...,a 1 ,a 0 )of elements from F q ,witha n ≠0.Thenumbern is the degree of the polynomial. Theorem 2.8 (Lagrange’s Interpolation) Let x 0 ,x 1 ,...,x d be distinct elements of a field F q and y 0 ,y 1 ,...,y d be any elements of F q . There exists a unique polynomial p(x) over F q with degree ≤ d such that p(x i )=y i for 1 ≤ i ≤ n. Algorithm 2.4 ( Interpolation(x 0 ,x 1 ,...,x d ,y 0 ,y 1 ,...,y d ) ) 1: return ⎛ ⎜ ⎝ 1 x 0 ... x d 0 1 x 1 ... x d 1 . . . .. . 1 x d ... x d d ⎞ ⎟ ⎠ −1 ⎛ ⎜ ⎝ ⎞ y 0 y 1 ⎟ . ⎠ y d Adi Shamir Of course the matrix inversion is to be performed over F q ,whichmeansall additions, subtractions and multiplications are calculated within the field, and divisions are performed by multiplying with the multiplicative inverse in the field.
Suppose Alice wants to distribute a secret S among n people P 1 ,P 2 ,...,P n in such a way that any k of them can recover the secret from their joint information, while it remains perfectly secret when any k − 1orlessofthem get together. This is what we call a [n, k]-secret sharing scheme. Algorithm 2.5 ( SSSS(S) ) 1: a 0 ← S, 2: FOR i := 1 TO k − 1 DO a i ← uniform(0..p − 1) 3: FOR j := 1 TO n DO s i ← a k−1 j k−1 + ...+ a 1 j + a 0 mod p 4: RETURN s 1 ,s 2 ,...,s n .
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COMP-547A /B Cryptography and Data
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Euclid Leonhard Euler Greatest Comm
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1.2.1 Fast modular exponentiation T
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1.2.2 GCD calculations and multipli
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EXAMPLE gcd(7,19) = gcd(19,7) = gcd
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EXAMPLE gcd(7,19) = gcd(1,0) = 1 x
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EXAMPLE 7x ≡ 5 (mod 9) 7 -1 ∙7x
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Finally, we know that a solution x
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1.3.1 Chinese Remainder Theorem Qin
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1.4 Quadratic Residues Quadratic re
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Properties ( 1 =+1 n) ( ) ab ( a )
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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+
Page 73 and 74: EXTRA SLIDES
Page 75 and 76: First, consider the case e =2. Sinc
Page 77: Second, notice that the same exact
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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