Re: [math-fun] Space for Sieve of Eratosthenes ~ O(n^(1/3))
So at some increase in complexity, this could be (theoretically) improved through increased fragmentation? What is the theoretical lower space bound on sieving -- if known? (Does a lower space bound even make any sense relative to brute force?) At 08:57 AM 9/27/2016, Tomas Rokicki wrote:
From the professor's comment on slashdot, it appears it's an incremental sieve that works on the span [N, N+N^(1/3)] and, rather than consider all potential prime factors < N^(1/2), it figures out which potential prime factors cannot possibly have multiples in the range [N, N+N^(1/3)] (which is the great majority).
-tom
On Tue, Sep 27, 2016 at 8:51 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Can anyone provide a 30-second explanation of this algorithm?
https://science.slashdot.org/story/16/09/26/2128254/ researcher-modifies-sieve-of-eratosthenes-to-work-with-less-physical-memory-space
Researcher Modifies Sieve of Eratosthenes To Work With Less Physical Memory Space (scientificamerican.com)
Posted by BeauHD on Monday September 26, 2016 @07:30PM from the ones-and-zeros dept.
grcumb writes: Peruvian mathematician Harald Helfgott made his mark on the history of mathematics by solving Goldbach's weak conjecture, which states that every odd number greater than 7 can be expressed as the sum of three prime numbers.
Now, according to Scientific American, he's found a better solution to the sieve of Eratosthenes:
"In order to determine with this sieve all primes between 1 and 100, for example, one has to write down the list of numbers in numerical order and start crossing them out in a certain order: first, the multiples of 2 (except the 2); then, the multiples of 3, except the 3; and so on, starting by the next number that had not been crossed out. The numbers that survive this procedure will be the primes. The method can be formulated as an algorithm."
But now, Helfgott has found a method to drastically reduce the amount of RAM required to run the algorithm:
"Now, inspired by combined approaches to the analytical 100-year-old technique called the circle method, Helfgott was able to modify the sieve of Eratosthenes to work with less physical memory space. In mathematical terms: instead of needing a space N, now it is enough to have the cube root of N."
So what will be the impact of this?
Will we see cheaper, lower-power encryption devices?
Or maybe quicker cracking times in brute force attacks?
Mathematician Jean Carlos Cortissoz Iriarte of Cornell University and Los Andes University offers an analogy:
"Let's pretend that you are a computer and that to store data in your memory you use sheets of paper. If to calculate the primes between 1 and 1,000,000, you need 200 reams of paper (10,000 sheets), and with the algorithm proposed by Helfgott you will only need one fifth of a ream (about 100 sheets)," he says.
http://www.scientificamerican.com/article/new-take-on-an-ancient-method-impr...
-- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
I'm not sure when you have to stop calling it a sieve, but if the pseudosquare sieve counts that's quite a bit smaller -- O(N^0.132) unconditionally, O((log N)^2) on a conjecture stronger than ERH. Charles Greathouse Case Western Reserve University On Tue, Sep 27, 2016 at 12:08 PM, Henry Baker <hbaker1@pipeline.com> wrote:
So at some increase in complexity, this could be (theoretically) improved through increased fragmentation?
What is the theoretical lower space bound on sieving -- if known? (Does a lower space bound even make any sense relative to brute force?)
At 08:57 AM 9/27/2016, Tomas Rokicki wrote:
From the professor's comment on slashdot, it appears it's an incremental sieve that works on the span [N, N+N^(1/3)] and, rather than consider all potential prime factors < N^(1/2), it figures out which potential prime factors cannot possibly have multiples in the range [N, N+N^(1/3)] (which is the great majority).
-tom
On Tue, Sep 27, 2016 at 8:51 AM, Henry Baker <hbaker1@pipeline.com> wrote:
FYI -- Can anyone provide a 30-second explanation of this algorithm?
https://science.slashdot.org/story/16/09/26/2128254/ researcher-modifies-sieve-of-eratosthenes-to-work-with- less-physical-memory-space
Researcher Modifies Sieve of Eratosthenes To Work With Less Physical Memory Space (scientificamerican.com)
Posted by BeauHD on Monday September 26, 2016 @07:30PM from the ones-and-zeros dept.
grcumb writes: Peruvian mathematician Harald Helfgott made his mark on the history of mathematics by solving Goldbach's weak conjecture, which states that every odd number greater than 7 can be expressed as the sum of three prime numbers.
Now, according to Scientific American, he's found a better solution to the sieve of Eratosthenes:
"In order to determine with this sieve all primes between 1 and 100, for example, one has to write down the list of numbers in numerical order and start crossing them out in a certain order: first, the multiples of 2 (except the 2); then, the multiples of 3, except the 3; and so on, starting by the next number that had not been crossed out. The numbers that survive this procedure will be the primes. The method can be formulated as an algorithm."
But now, Helfgott has found a method to drastically reduce the amount of RAM required to run the algorithm:
"Now, inspired by combined approaches to the analytical 100-year-old technique called the circle method, Helfgott was able to modify the sieve of Eratosthenes to work with less physical memory space. In mathematical terms: instead of needing a space N, now it is enough to have the cube root of N."
So what will be the impact of this?
Will we see cheaper, lower-power encryption devices?
Or maybe quicker cracking times in brute force attacks?
Mathematician Jean Carlos Cortissoz Iriarte of Cornell University and Los Andes University offers an analogy:
"Let's pretend that you are a computer and that to store data in your memory you use sheets of paper. If to calculate the primes between 1 and 1,000,000, you need 200 reams of paper (10,000 sheets), and with the algorithm proposed by Helfgott you will only need one fifth of a ream (about 100 sheets)," he says.
http://www.scientificamerican.com/article/new-take-on-an- ancient-method-improves-way-to-find-prime-numbers/
-- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
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