This turned out to be a bit verbose, but I hope it fully answers your question… A Gaussian integer is a complex number of the form G = a+bi where i2 = -1, and a and b are integers. The Gaussian integers form a unique factorization domain. Some of them act as units (e.g. 1, … Read more
From one of my favorite books ever, Applied Cryptography by Bruce Schneier “If someone created a database of all primes, won’t he be able to use that database to break public-key algorithms? Yes, but he can’t do it. If you could store one gigabyte of information on a drive weighing one gram, then a list … Read more
The basic idea of a segmented sieve is to choose the sieving primes less than the square root of n, choose a reasonably large segment size that nevertheless fits in memory, and then sieve each of the segments in turn, starting with the smallest. At the first segment, the smallest multiple of each sieving prime … Read more
RSA doesn’t pick from a list of known primes: it generates a new very large number, then applies an algorithm to find a nearby number that is almost certainly prime. See this useful description of large prime generation): The standard way to generate big prime numbers is to take a preselected random number of the … Read more
If you don’t want to reinvent the wheel, use the library sympy pip install sympy Use the function sympy.ntheory.factorint Given a positive integer n, factorint(n) returns a dict containing the prime factors of n as keys and their respective multiplicities as values. For example: Example: >>> from sympy.ntheory import factorint >>> factorint(10**20+1) {73: 1, 5964848081: … Read more
Here’s the best algorithm I know of (in Python) def prime_factors(n): “””Returns all the prime factors of a positive integer””” factors = [] d = 2 while n > 1: while n % d == 0: factors.append(d) n /= d d = d + 1 return factors pfs = prime_factors(1000) largest_prime_factor = max(pfs) # The … Read more