[Crypto] Prime factorization (102 digits)

Solution 1:

Your 102-digit nuber is two digits more than the first RSA challenge RSA-100 that has 330-bit.

This can be easily achieved with existing libraries like;

  • CADO-NFS ; http://cado-nfs.gforge.inria.fr/

  • NFS factoring: http://gilchrist.ca/jeff/factoring/nfs_beginners_guide.html

  • Factoring as a service https://seclab.upenn.edu/projects/faas/

    The Factoring as a Service project is designed to allow anyone to factor 512-bit integers in as little as four hours using the Amazon EC2 platform for less than $100, with minimal setup.

The experiment

Factor of a 99 digit number $n =$ 112887987371630998240814603336195913423482111436696007401429072377238341647882152698281999652360869.

  1. I have tried with Pollard's $p$ -1 algorithm, still running for one and a half-day and did not produce a result, yet. This is what expected due to the B bound must be around $2^{55}$ with success probability $\dfrac{1}{27}$. I've stopped the experiment after the CADO-NFS succeeds. This is self-implemented Pollard's $p$ -1, one can find an optimized in GMP-ECM

  2. Tried the CADO-NFS. The stable version may not be easily compiled for new machines, so prefer the active one from the GitLab.

    After ~6 hours with 4 cores, CADO-NFS produced the result. This was an RSA CTF/Challange and I don't want to spoil the fun; here the hash commitments with SHA-512, it is executed with OpenSSL;

    echo -n "prime x" | openssl sha512



Experiments on RSA challenges with 6 cores using CADO-NFS

RSA Challange Bit size Time in minutes
RSA-100 330 270
RSA-110 364 280
RSA-120 397 1049
RSA-129 426 3279
RSA-140 430 Not tested

enter image description here The core count is very important to reduce the time as 512-bit can be broken as 4 hours in the EC2 platform.

Details of the experiment

  • CPU : Intel(R) Core(TM) i7-7700HQ CPU @ 2.80GHz

  • RAM : 32GB - doesn't require much ram, at least during polynomial selection and Sieveing.

  • Dedicated cores : 4

  • Test machine Ubuntu 20.04.1 LTS

  • CUDA - NO

  • gcc version 9.3.0 (Ubuntu 9.3.0-17ubuntu1~20.04)

  • cmake version 3.16.3

  • external libraries: Nothing out of Ubuntu's canonicals

  • CODA-NFS version : GitLab develepment version cloned at 23-01-2021

  • The bit sizes;

    • $n$ has 326 bits
    • $p$ has 165 bits
    • $q$ has 162 bits

The cado-nfs-2.3.0 did not compile and giving errors about HWLOC- HWLOC_TOPOLOGY_FLAG_IO_DEVICES. Asked a friend to test the compile and it worked for them. It was an older Linux version. So I decided to use the GitLab version.

  • Note: this question did not factor the OPs original number.

  • Historical note: RSA-100 challenge has 330 bits and broken by Lenstra in 1991.

Solution 2:

Using for example cado-nfs, you can find the factorization (~5min using 32 cores) as 51700365364366863879483895851106199085813538441759 * 3211696652397139991266469757475273013994441374637143