add lenstra eliptic curve factoring

[oscardssmith]

heavily inspired by @trizen, but refactored a little to make it more reliable.

[oscardssmith]

some benchmarks:

x = prod(nextprime(big(2)^30, i) for i in 1:2)
@time Primes.pollardfactors!(x, Dict{BigInt,Int}())
  0.028354 seconds (1.13 M allocations: 22.163 MiB)
@time lenstrafactors!(x, Dict{BigInt,Int}())
  0.110741 seconds (1.65 M allocations: 30.821 MiB, 26.82% gc time)

x = prod(nextprime(big(2)^30, i) for i in 1:20)
@time Primes.pollardfactors!(x, Dict{BigInt,Int}());
  0.830832 seconds (14.86 M allocations: 346.059 MiB, 21.69% gc time)
@time lenstrafactors!(x, Dict{BigInt,Int}());
  0.375128 seconds (3.62 M allocations: 114.038 MiB, 11.86% gc time)

x = prod(nextprime(big(2)^50, i) for i in 1:2)
@time lenstrafactors!(x, Dict{BigInt,Int}())
 18.921172 seconds (296.05 M allocations: 6.041 GiB, 14.56% gc time)
@time Primes.pollardfactors!(x, Dict{BigInt,Int}())
  56.697309 seconds (1.12 G allocations: 21.651 GiB, 20.29% gc time)

These are the bad cases for ecm (when all the factors are of similar size). That said, ecm tends to do a bunch better for larger inputs, and for numbers with a few smaller factors. This implementation is not especially optimized, but I think it is a good baseline for future improvements.

[codecov]

Codecov Report

❌ Patch coverage is 97.11538% with 3 lines in your changes missing coverage. Please review.
✅ Project coverage is 94.97%. Comparing base (b2fca94) to head (42f0316).

Files with missing lines	Patch %	Lines
src/Primes.jl	97.11%	3 Missing ⚠️

Additional details and impacted files

@@            Coverage Diff             @@
##             main     #104      +/-   ##
==========================================
+ Coverage   93.52%   94.97%   +1.45%     
==========================================
  Files           2        2              
  Lines         463      517      +54     
==========================================
+ Hits          433      491      +58     
+ Misses         30       26       -4     

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[oscardssmith]

With a little help from claude, I now have this working a lot faster (due to a better basis for curve and adding stage 2).

min factor	pollard	lenstra stage-1-only	lenstra full	stage-2 speedup
20-bit	36 µs	121 µs	149 µs	0.81×
25-bit	159 µs	281 µs	385 µs	0.73×
30-bit	950 µs	350 µs	518 µs	0.68×
35-bit	425 ms	139 ms	65 ms	2.15×
40-bit	1.67 s	107 ms	60 ms	1.76×
45-bit	—	995 ms	265 ms	3.76×
50-bit	—	2.81 s	680 ms	4.13×
55-bit	—	4.86 s	2.00 s	2.43×

[oscardssmith]

@s-celles tagging you here cause I figure you may be interested.