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Integer Factorizations

Trial division by all integers up to B to factor number up to B²
Pollard-Rho factors numbers up to B⁴ (usually)
- Works well in practice on numbers with small factors
- Analysis: $\Theta(\sqrt{p})$ to find factor p
  To factor $\beta$ -bit composite number n
  Try all prime factors $< \;\lfloor \sqrt{n} \rfloor$
  The run time is $n^{\frac{1}{4}} \;=\; 2^{\frac{\beta}{4}}$ arithmetic operations
  $n^{\frac{1}{4}} \beta^3 \;=\; 2^{\frac{\beta}{3}}$ bit operations

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