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fastgcd
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Nadia Heninger
J. Alex Halderman

Fastgcd is a C implementation of an efficient algorithm to compute the pairwise
GCDs of a collection of integers.  The algorithm is due to Bernstein and is a
simplified version of (http://cr.yp.to/papers.html#smoothparts).

The input to the program is a file listing the collection of integers (e.g.,
RSA moduli).  The algorithm will compute the GCD of each input integer with the
product of every other input integer, and output the nontrivial common divisors
along with a list of input integers which had a nontrivial common divisor.  In
the simplest case for RSA moduli, the common divisor will be a single prime
factor.  We leave it as an exercise to deal with cases where the divisor is
composite.

This program was used in "Mining your Ps and Qs: Detection of Widespread Weak
Keys in Network Devices" to discover weak RSA keys. For more information, and a
copy of the paper, see https://factorable.net/


Install
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To install fastgcd (in current directory), simply run

./install.sh

The install script will download and install a local copy of gmp-5.0.5 and
extract and apply our patch to GMP.  This patch modifies GMP to add an integer
output format that can accommodate integers larger than 2^32 bytes, which is
necessary for large inputs to the algorithm.  This means that the temporary
files written to disk in the course of the algorithm by our patched GMP are not
compatible with the standard GMP output format.

Running
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To run a fresh instance of fastgcd, use

./fastgcd input.moduli

On completion, fastgcd will output:

vulnerable_moduli       - list of moduli that had a nontrivial common divisor
                          with any input modulus, in hex, one per line.

gcds                    - list of the common divisor of each modulus in 
                          vulnerable_moduli with the product of every other 
                          modulus in input.moduli, in hex, one per line.


We have provided a sample input:
input.moduli            - list of moduli, in hex, one per line. Note: these
                          must be unique.

Sample output produced by running fastgcd on the provided input.moduli file are
contained in sample.out/.