On Approximating Addition by Exclusive OR

@article{Sarkar2009OnAA,
  title={On Approximating Addition by Exclusive OR},
  author={Palash Sarkar},
  journal={IACR Cryptology ePrint Archive},
  year={2009},
  volume={2009},
  pages={47}
}
Let X, X, . . . , X be independent and uniformly distributed over the non-negative integers {0, 1, . . . , 2 − 1}; S = X +X + · · ·+X and L = X ⊕X ⊕ · · · ⊕X. Denote the i-th bits of S and L by S i and L (n) i respectively. We show that as i→∞, Pr[S (n) i = L (n) i ]→ γ = 1 2 + 2n+1(2n+1−1) 2(n+1) × bn+1 n! , where bn is the n-th Bernoulli number. As a consequence, γ (2r) = 1/2 for every r; and we show that γ → 1/2 as r → ∞. For small values of r, γ is significantly different from 1/2; for… CONTINUE READING

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