  We describe a new algorithm for computing Whitney stratifications of complex
projective varieties. The main ingredients are (a) an algebraic criterion, due
to L\^e and Teissier, which reformulates Whitney regularity in terms of
conormal spaces and maps, and (b) a new interpretation of this conormal
criterion via primary decomposition, which can be practically implemented on a
computer. We show that this algorithm improves upon the existing state of the
art by several orders of magnitude, even for relatively small input varieties.
En route, we introduce related algorithms for efficiently stratifying affine
varieties, flags on a given variety, and algebraic maps.
