A factorization witness for n only needs to list factors once each; even with many small primes, encoding k factors takes O(k log n) bits, which is polynomial in log n when k is bounded by log n.

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Bounded (bounded by log n)(in mathematics): Limited or capped at a certain maximum value; here it means the number of factors k doesn't exceed the logarithm of n.
Factorization witness(in computational complexity and number theory): A list or proof that shows what prime numbers multiply together to make a larger number; it's evidence that you've correctly broken down a number into its basic building blocks.
O(k log n) notation (Big O notation)(in computer science and computational complexity): A way to describe how much computer memory or time something needs as the input gets bigger; O(k log n) means the amount grows roughly proportionally to k times the logarithm of n.
Polynomial time/Polynomial in log n(in computational complexity theory): A measure of computational efficiency; an algorithm runs in polynomial time if the resources it needs grow at a reasonable, controlled rate as the input gets bigger.

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