Dixon's Factorization Method - Example

Example

We will try to factor N = 84923 using bound B = 7. Our factor base is then P = {2, 3, 5, 7}. We then search randomly for integers between and N whose squares are B-smooth. Suppose that two of the numbers we find are 513 and 537:

So

Then

\begin{align} & {} \qquad (513 \cdot 537)^2 \mod 84923 = (275481)^2 \mod 84923 \\ & = (84923 \cdot 3 + 20712)^2 \mod 84923 \\ & =(84923 \cdot 3)^2 + 2\cdot(84923\cdot 3 \cdot 20712) + 20712^2 \mod 84923 \\ & = 0 + 0 + 20712^2 \mod 84923 \end{align}

That is,

The resulting factorization is 84923 = gcd(20712 − 16800, 84923) × gcd(20712 + 16800, 84923) = 163 × 521.

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