Concurrent Rho algorithm + Smart-contract

Implementation of Pollard's Rho algorithm in Go. Also check out smart-contract for Ethereum written in Solidity(see ethereum/contracts/Rho.sol) and sequential Python implementation(see rho.py).

How it works

Rho algorithm is generating the sequence of numbers x0, x1 = f(x0), ..., x_i = f(x_{i-1}), where f(x) = x*x + c. We are applying Floyd's cycle detection algorithm, to find point in the cycle and this can give us factor. For more details see wiki.

We make it concurrent by running with different starting parameters x0 and c. As soon as one of the subroutines find factor, we stop all the other. This simple concurrent modification of algorithm can give us performance gain.

Usage

To run sequential version simply call algo.Factorize(n). If you want to use concurrent implementation, you'll need to specify additional parameter j which is concurrency limit. Eg. algo.FactorizeParallel(n, 4).

To run benchmarks just do:

$ go test -bench . -benchtime 1m -timeout 1h ./...

Results

As you know, Rho algorithm belongs to randomized, so it's hard to get accurate performance stats.

Here are a couple of my empirical observations:

• There is number 4 that algorithm is failing to factorize.
• For small numbers(eg. 8, 16, 25) concurrent solution can find factor faster. These numbers have high probability of cycling in Rho procedure. Only some specific pairs (x0, c) lead to the result.
• For numbers(even very big) that have many small factors, algorithm will find answer very quick. Parallel execution here will not help.
• For numbers with only a few factors, concurrent algorithms can help.

Here are benchmark results from running program on my laptop(Intel Core i7-6600U @ 4x 3.4GHz).

Name	Runs	Time(ms/op)
Parallel_6_1	10000	13.240
Parallel_6_2	10000	13.176
Parallel_6_3	10000	13.681
Parallel_6_4	5000	16.099
Parallel_7_1	2000	54.945
Parallel_7_2	2000	57.374
Parallel_7_3	2000	53.736
Parallel_7_4	2000	63.402
Parallel_8_1	500	201.967
Parallel_8_2	500	205.176
Parallel_8_3	500	194.191
Parallel_8_4	300	243.370
Parallel_9_1	50	2567.833
Parallel_9_2	50	2264.897
Parallel_9_3	30	2317.177
Parallel_9_4	50	2005.279

Where each benchmark is based on prod prime[i] for i from Left to Right.

Name	Left	Right
Parallel_6	10^6	10^6 + 5
Parallel_7	10^7	10^7 + 5
Parallel_8	10^8	10^8 + 5
Parallel_9	10^9 + 1	10^9 + 5

And last part of benchmark name states for concurrency limit used for the test.

Running on Ethereum

There is also an implementation of the Pollard's Rho algorithm as a smart-contract for Ethereum. You can find it in ethereum/contracts/Rho.sol. It's able to find a factor of 256bit integer. Here is how you can test it using Truffle Framework.

Install

$ sudo npm install -g truffle

Run

Let's run development blockchain built into Truffle

$ cd ethereum
$ truffle develop

This should give you a prompt like truffle(develop)>.

Now we need to compile and publish our contract

truffle(develop)> compile
Compiling ./contracts/Migrations.sol...
Compiling ./contracts/Rho.sol...
Writing artifacts to ./build/contracts

truffle(develop)> migrate
Using network 'develop'.
...

After this we can use our smart contract like this

truffle(develop)> var rho = Rho.at(Rho.address)
truffle(develop)> var x0 = 2, p = 1, n = 35
truffle(develop)> rho.run(x0, p, n)
BigNumber { s: 1, e: 0, c: [ 7 ] }

Result is a BigNumber representation of our factor. In our case factor 7 is in field called c.

Computation cost

You can try to predict gas used for the factorization by calling estimateGas function

truffle(develop)> rho.run.estimateGas(2, 1, 169)
28989

And of course when you run it, you can specify gas limit and gas price

truffle(develop)> rho.run(2, 1, 169, {gas: 30000, gasPrice: web3.fromWei(1, 'wei')})
BigNumber { s: 1, e: 1, c: [ 13 ] }

This is what happens if you give not enough gas

truffle(develop)> rho.run(2, 1, 169, {gas: 28000})
Error: Error: VM Exception while executing eth_call: out of gas
...