[WIP] Add an example of generating large primes

[cuviper]

This example generates large BigUints, then tests for primality.

This is marked WIP because I see a few ways I hope to make this faster by making is_prime part of the API, and improving the implementation with internal details -- like leveraging modpow's Montgomery domain more.

[cuviper]

My attempts to stay in the Montgomery domain haven't made any difference. The bottleneck is in the actual exponentiation. But I think the first witness of 2 could do a lot better optimized as shifts instead. It's something to try, anyway...

[cuviper]

Meh. I tried shifting the base up to the size of the modulus, but for N-bit exponent and modulus, this only reduces the number of iterations by log N.

[dignifiedquire]

hey, this might be of interest to you. I am implementing rsa in rust using this library, porting straight from the golang stdlib as a starting point. I have ported the prime tests already, and getting these numbers.

Source: https://github.com/dignifiedquire/rust-rsa/blob/master/src/prime.rs
Golang: https://github.com/golang/go/blob/master/src/math/big/prime.go

The benchmarks are a 1-1 port, using the same numbers so results should be comparable well.
Note1: golang has an optimized big implementation using a lot of assembler
Note2: I haven't gone through optimizing my code yet, just straight porting and making sure tests are passing, so likely I am missing some low hanging fruit there.

Rust

test bench_prime_lucas        ... bench:  11,991,982 ns/iter (+/- 626,235)
test bench_prime_miller_rabin ... bench:   3,195,908 ns/iter (+/- 347,849)
test probably_prime_0         ... bench:  14,820,231 ns/iter (+/- 1,600,128)
test probably_prime_1         ... bench:  18,727,275 ns/iter (+/- 2,883,921)
test probably_prime_10        ... bench:  48,884,033 ns/iter (+/- 4,318,378)
test probably_prime_20        ... bench:  77,457,939 ns/iter (+/- 1,612,339)
test probably_prime_5         ... bench:  30,915,429 ns/iter (+/- 2,710,209)

golang

BenchmarkProbablyPrime/n=0-4         	     500	   3778513 ns/op
BenchmarkProbablyPrime/n=1-4         	     300	   4386977 ns/op
BenchmarkProbablyPrime/n=5-4         	     200	   6867582 ns/op
BenchmarkProbablyPrime/n=10-4        	     100	  10186901 ns/op
BenchmarkProbablyPrime/n=20-4        	     100	  16642978 ns/op
BenchmarkProbablyPrime/Lucas-4       	     300	   3692549 ns/op
BenchmarkProbablyPrime/MillerRabinBase2-4         	    2000	    677351 ns/op

name	go	rust
lucas	3,692,549 ns/op	11,991,982 ns/iter
miller_rabin	677,351 ns/op	3,195,908 ns/iter
n_0	3,778,513 ns/op	14,820,231 ns/iter
n_1	4,386,977 ns/op	18,727,275 ns/iter
n_5	6,867,582 ns/op	30,915,429 ns/iter
n_10	10,186,901 ns/op	48,884,033 ns/iter
n_20	16,642,978 ns/op	77,457,939 ns/iter

PS: using a fork as I am missing access to raw limbs, trailing zeros and the Roots impl currently
PPS: if you have time and have ideas on how I can optimize my code, it would be very much appreciated!