How to efficiently compute a batch GCD posted December 2015
Heard a bit late about the factorable research results and how they used batch gcd to recover a bunch of servers' private keys.
The question one could think of is how to efficiently do a batch gcd on a big set of public keys?
From this utility:
- Actual pairwise GCD
This performs n*(n-1)/2 GCD operations on the moduli. This is slow. Don't use this.
- Accumulating Product
This iterates over all input moduli, performing a GCD of each one against the product of all previous. Once it finds a candidate, it scans all previous moduli to find out which ones it shared a factor with (either GCD or division, depending on whether one or both were found). The main scan cannot be done in parallel, and even though it seems like this is O(n), the increasing size of the accumulated product results it lots of long multiplication and long divison so it's still painfully slow for large numbers of moduli.
- Smooth Parts
DJB's "How to find smooth parts of integers" http://cr.yp.to/papers.html#smoothparts This creates a product tree, then converts it to a remainder tree, then kablam you find common factors. This is largely the same as the one at https://factorable.net/resources.html This is the default, use this unless you're trying to burn in a new CPU or something.
Looks like the most efficient ways come from Dan Bernstein (again!), in a 7 pages paper