Re: Public Key, Symbolic Calculation

websnarf_at_gmail.com
Date: 02/09/05 

Date: 8 Feb 2005 16:24:26 -0800

Kiuhnm wrote:
> tomstdenis@gmail.com wrote:
> > What does "algebraic coefficients" mean anyways? The coefficients
of a
> > polynomial must belong to a group, ring or field to be
representable.
> > I'm afraid I don't know what group "algebraic" is... [punk or ska?]
>
> You only need integers.
> For example:
> sin(sqrt(345093/41 + (345313)^(1/5) - (45645)^(1/3)))^2

Ack! No! That's not what *I* meant by "Algebraics". Sorry, if I am
not using the most precise terminology. What *I* meant by "algebraics"
was the root of any polynomial with coefficients in Z. But even
further than that, I can restrict this to those solutions which are
real (I don't know if that's necessary or desirable, but my point was
not introduce complex numbers if they didn't add anything useful). I
am pretty sure that creates a "field" (though admittedly, I can't prove
it off the top of my head).

I definately, had no intention of introducing "sin" or other
transcendentals into there. In fact I don't believe adding
transcendentals would actually help in the security of this (remember,
he wanted everything in symbolic notation -- there will be little or no
collapsing of those transcendentals; so it would actually probably
leave easy breadcrumbs for directly reverse engineering what the roots
were in the first place.)

Also I want to make clear that I see the possibility of developing an
idea here -- I don't see this as anything resembling a real "method"
just yet. But Abel's statement that there is no deterministic way of
factoring polynomials, seems a little stronger than our current
*BELIEF* that we cannot factor integers in a deterministic way. So
there might be something here that is worth pursuing. 

--
Paul Hsieh
http://www.pobox.com/~qed/
http://bstring.sf.net/

Relevant Pages

  • Re: JSH: Heart of dispute, number properties
    ... >> with irrationals, where one root is like the integer root before, ... algebraic integer A, there exists another algebraic integer B ... of some kind in this ring, perhaps even an algebraic integer, and other ... taste, i.e., his feeling that polynomials should factor in a different ...
    (sci.math)
  • Re: Public Key, Symbolic Calculation
    ... >> polynomial must belong to a group, ring or field to be ... was the root of any polynomial with coefficients in Z. But even ... transcendentals would actually help in the security of this (remember, ... factoring polynomials, seems a little stronger than our current ...
    (sci.crypt)
  • Re: Possible flaw in the polynomial ring A[X] construction
    ... distinguish it from the product of polynomial functions. ... for rings of finite characteristic, so let us consider the base ring to ... the ring of polynomials over this field is ... with the base ring being the field of two elements. ...
    (sci.math)
  • Re: Possible flaw in the polynomial ring A[X] construction
    ... with further operations such as a quotient ring. ... then the ring Aof polynomials with real coefficients, ... than just heaping piles of construction upon other piles of ... "While the polynomial itself is necessarily infinite in length, ...
    (sci.math)
  • Re: Understanding the quotient ring nomenclature
    ... A ring is defined as containing elements. ... undefined domain and an infinite length of terms. ... infinite series are reals. ... long as the two polynomials being multiplied have finitely many terms, ...
    (sci.math)
Quantcast