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Registered Member #2099
Joined: Wed Apr 29 2009, 12:22AM
Location: Los Altos, California
Posts: 1714
As many of us know superficially, lots of public key cryptography is based on the computational difficulty of factoring large numbers. I've never studied that, but maybe other readers of this forum have. Anyone got some good references about integer arithmetic with numbers larger than the computer's natural word size?
Following a challenge by Weston in another thread, I looked up some "sort of big" prime numbers and computed their product: 944,871,244,210,099,859. Then got to thinking, that's not big enough. I bet it could be factored within minutes with an ordinary program on an ordinary computer, written in less than an hour. Anybody want to try that, or show us an online calculator that can do the job?
Then took it to the next level by making the product bigger than 64 bits. Just to separate the men from the boys. (Toddlers from crawlers would be a better analogy). Here's a bigger product of two primes: 1,030,935,892,468,536,050,027. Huge hint: I copied the factors from a familiar web page. If I were challenged to factor that, the trade off would be an hour on Internet searching for a factoring tool, vs. an hour or two writing and benchmarking a primitive factoring program (start with the sub-64 bit problem).
Is anyone else interested in this stuff, but not enough to go factoring yourself? Then I could post one factor, and the much simpler exercise will be to find the other. So far I've done all this work in Excel, using base-1000 or base-10000 arithmetic.
I bet most bitcoin mining is done by people who aren't programmers or mathematicians.
Registered Member #61406
Joined: Thu Jan 05 2017, 11:31PM
Location:
Posts: 268
RSA Rc4/5 convert bytes to bits and store it in array, then do binary maths, in there code. I think just trying to divide a number not with itself or one, and if there's always a carry bit, its a prime. Divide and mulpilcation just loop add or subtract, I think, not sure if there's a short cut.
Registered Member #235
Joined: Wed Feb 22 2006, 04:59PM
Location:
Posts: 80
You would be surprised at the level of skill for some of the first miners. Now you can just buy an ASIC to do the work for you. Most of the public key cryptography is based on elliptic curve cryptography. The issue with finding the private key is the arithmetic is modulo based. Bitcoin uses 256bit modulo arithmetic, in as such the result wraps around many many times. Finding the original private key is solving discrete logarithm problem in O time. Quatum computers could do it with Shor's algorithm.
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large integer arithmetic
