Quadratic sieve implementation in Java
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Ilya Gazman
Latest commit 76fdb1d Oct 27, 2017

README.md

Java Quadratic Sieve

This is Quadratic sieve implementation in Java, you can find a lot of info about the algorithm at wiki, so I just assume you are familiar with the basics of the algorithm and jump right in to the implementation.

news

Upgraded to version 3:

  • Removed the mod from the sieve loop by replacing it with if bigger then do -, it improved performance by 20%
  • Removed the mode from the wheels, this one was more tricky as I had to add a new state for that, the gain was 50% performance
  • Changed Double to double, it improved performance by 500%, that one was funny
  • Added support for multithreaded environment, each thread works on it's own region of numbers, it adds about 70% gain in performance for each core you got
  • Moved to Java Gradle with Java 8

After the improvements the algorithm can handle 160 bits(49 digits) in ~40 sec

Implementation

  • First I start by building the prime base. I am using a predefined B_BOUNDS value to determinate how many primes I want to include in it. I iterate over all the numbers starting from 2, and using the primality test to check if they are prime(BigInteger.nextProbablePrime). I am using Legendre symbol to include only those primes that are in quadratic residue mod N.
  • Next I am building wheels, each wheel holds one prime, it tells what is the next number that the prime will divide. I use them during the sieving loop. In order to find the first tow values of each prime I use Tonelli–Shanks algorithm, he called it ressol. Also the wheel calculates the prime log. This is in order to avoid unnecessary division during the sieve loop.
  • Now I star the sieve loop, I start checking values from the ceil root of N(the values that I am trying to factor) and define sieveVectorBound as the size of the biggest prime in the factory base, this is how many numbers I will process over each loop. I choose this size so each wheel will be used attlist once.
    • First I calculate the base log, it's the size ceil root of N plus current position, position stats from 0 and increased by sieveVectorBound each loop.
    • I create 2 arrays of doubles, logs and trueLogs with the size of sieveVectorBound.
    • I create another array of VectorData with the size of sieveVectorBound. VectorData contains information about the position where it been found, and the b-smooth vector already transformed to boolean, I use BitSet for it. Position is used to calculate x and y such that x^2 - N = y, it's important for the last step of the algorithm.
    • Now I iterate over all the wheels and sum all their logs in the logs array. I do it with the help of tow important methods that wheel got:
      • testMove(long limit) it checks if the wheel already reached a limit position
      • nextLog() it increase the wheel current position and return the log for that position(At this point its the same log all the time, but in early implementations I been returning the log multiplied by the biggest power of that prime that could divide that position)
    • Next I return the wheels to the position they been at when this loop started and make another iteration over them all
    • For each wheel I check if it's corresponding log[index] is bigger than the log of biggest prime in the prime base power tow. If it's bigger I skip to the next wheel but if it's not, this number is aether a prime or divided by one of the prime base primes.
    • So if not, I find the actual log of that position by taking it from the trueLogs array or calculating and saving it there if it's the first time. I compare the actual log to the log I calculated if its equal withing a small error of 0.0000001, I know that this position is fully factored by the prime base and I add its VectorData to bSmoothVectors VectorData array to be used latter.
    • if its not equal than it's a prime, so I add its VectorData to bigPrimesList.
    • We are still in the if case of not ignoring the wheel, we added the VectorData to the right list and now I turn on the bit of VectorData b-smooth vector at the current index of the wheels array
  • When enough big primes and b-smooth vectors been found, I construct and solve the matrix using gaussian elimination, I know it can be improved by using algorithms to find the null space, but I am using a very efficient implementation with bit calculation, so it's not the bottle neck of the algorithm
  • I make small optimization in the bigPrimeList, I only take those big primes that been found more than once, and if the big prime been found just twice, I xor, those two vectors to one, as it's the only way it will be part of the solution.

After solving the matrix, I extract the solution and it's pretty much it. As of version 2.1 I can factor 140 bits numbers(40 digits) in half minute. The matrix solver can handle matrices of the size 100,000 in 15 minutes.

How can you help?

This code needs better documentation, the implementation description can be improved as well, I did it because I love math. I will appreciate any bug reports, comments and new ideas.