#19814: Number Theory::random_prime(n) generating composite numbers for large
values of n
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Reporter: DayneSorvisto | Owner:
Type: defect | DayneSorvisto
Priority: major | Status: new
Component: number theory | Milestone: sage-7.0
Keywords: prime numbers | Resolution:
Authors: | Merged in:
Report Upstream: Reported upstream. No | Reviewers:
feedback yet. | Work issues:
Branch: | Commit:
Dependencies: | Stopgaps:
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Description changed by DayneSorvisto:
Old description:
> Hi,
>
> First time posting a bug report on trac. The bug I'd like to report is
> "random_prime" returns composite numbers for large integers. For example
>
> {{{
> random_prime(100000000000000000000000000000000000000000000000000000000000000000000000000)
> }}}
>
> returned
> 71944724797690175157403755256108218794891687825722145298657347991279721529
> which is a composite number ending in 9. I am using the SageMath cloud
> notebook but believe it's reproducible on other implementations of Sage.
>
> Looking at the source in rings.arith.py it seems that the problem is this
> line which generates a probable prime for large numbers.
>
> {{{
> smallest_prime = ZZ(lbound-1).next_probable_prime()
> }}}
>
> Suggestions: Perhaps there could be some more error checking for large
> numbers so Sage doesn't return obviously composite numbers? Or at the
> very least checking small prime factors? If this function were being used
> in Cryptography, the Handbook of Applied Cryptography recommends the
> Miller Rabin test for random large primes.
New description:
Hi,
First time posting a bug report on trac. The bug I'd like to report is
"random_prime" returns composite numbers for large integers. For example
{{{
random_prime(100000000000000000000000000000000000000000000000000000000000000000000000000)
}}}
returned
71944724797690175157403755256108218794891687825722145298657347991279721529
which is a composite number ending in 9. I am using the SageMath cloud
notebook but believe it's reproducible on other implementations of Sage.
Looking at the source in rings.arith.py it seems that the problem is this
line which generates a probable prime for large numbers.
{{{
prime_test = is_pseudoprime
}}}
The is_prseudoprime function then generates the erroneous output.
According to the documentation this is an implementation of the Miller
Rabin Test and the Lucas Test.
Suggestions: Perhaps there could be some more error checking for large
numbers so Sage doesn't return obviously composite numbers? I suspect the
is_pseudoprime function doesn't check very small prime factors such as 3.
--
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Ticket URL: <http://trac.sagemath.org/ticket/19814#comment:7>
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