Beefy Boxes and Bandwidth Generously Provided by pair Networks
Perl-Sensitive Sunglasses
 
PerlMonks  

comment on

( #3333=superdoc: print w/replies, xml ) Need Help??
You're quite right, Fermat's method will only find two factors: it is important to note that all the fastest factoring algorithms (with the exception of the elliptic curve method) use this as a basis.

The Quadratic Sieve works as follows:

If n is the number to be factored, the QS tries to find x and y such that x^2 = y^2 (mod n) and x is not equal to plus or minus y; this implies that (x-y)(x+y)=0 and we compute GCD(x-y,n) to see if this is a non-trivial divisor. There is at least a 1/2 chance that the factor will be non-trivial. The first step is to define

Q(x)=(x+floor(sqrt(n)))^2-n=x'^2-n

and compute Q(x_1),Q(x_2),...,Q(x_k). Determining the x_i requires sieving over a factor base (using primes such that the Legendre symbol (n/p)=1). From the evaluations of Q(x) we want a subset Q(x_i1)Q(x_i2)...Q(x_ir) that is a square, y^2. Then note that for all x, Q(x)=x'^2 (mod n). So what we now have is that

Q(x_i1)Q(x_i2)...Q(x_ir)=(x_i1*x_i2*...*x_ir)^2 (mod n)

Which are the factors of n.


In reply to Re: Vampire Numbers by gumby
in thread Vampire Numbers by YuckFoo

Title:
Use:  <p> text here (a paragraph) </p>
and:  <code> code here </code>
to format your post; it's "PerlMonks-approved HTML":



  • Are you posting in the right place? Check out Where do I post X? to know for sure.
  • Posts may use any of the Perl Monks Approved HTML tags. Currently these include the following:
    <code> <a> <b> <big> <blockquote> <br /> <dd> <dl> <dt> <em> <font> <h1> <h2> <h3> <h4> <h5> <h6> <hr /> <i> <li> <nbsp> <ol> <p> <small> <strike> <strong> <sub> <sup> <table> <td> <th> <tr> <tt> <u> <ul>
  • Snippets of code should be wrapped in <code> tags not <pre> tags. In fact, <pre> tags should generally be avoided. If they must be used, extreme care should be taken to ensure that their contents do not have long lines (<70 chars), in order to prevent horizontal scrolling (and possible janitor intervention).
  • Want more info? How to link or How to display code and escape characters are good places to start.
Log In?
Username:
Password:

What's my password?
Create A New User
Domain Nodelet?
Chatterbox?
and the web crawler heard nothing...

How do I use this? | Other CB clients
Other Users?
Others drinking their drinks and smoking their pipes about the Monastery: (4)
As of 2023-01-29 16:52 GMT
Sections?
Information?
Find Nodes?
Leftovers?
    Voting Booth?

    No recent polls found

    Notices?