Re: RSA: more than one secret exponent d exists ???

Kristian Gjøsteen wrote:

Sorry. I'll try to sketch the argument below, with as little
mathematics as possible.
[...]

Still too complicated. Why not the easy way with some algebra?

The ring Z_n is a multiplicative composition of the mutually exclusive
rings Z_p and Z_q (with respect to multiplication). The relation between
e and d exists in Z_n, therefore must also exist in both Z_p and Z_q.
Now if m^(d*e)=m is true mod (p-1) and mod (q-1), you'll easily get that
is must be true mod lcm(p-1,q-1) as well.

This also shows that RSA can be applied to any finite fields and their
composition, where the decomposition of the composite is a well-known
hard problem.
.

Relevant Pages

  • Re: Is the polynomial ring interpretation of abstract algebra A[X]
    ... Addition and multiplication is defined between the elemens of that ring, ... functions of a base ring, as addition and multiplication are defined, this ... We can see these as polynomials too. ... composition of elements then there is an element which can represent ...
    (sci.math)
  • Re: Prove that a set is a monoid, but not a ring
    ... grimster wrote: ... >end nis not a ring? ... (under, presumably, pointwise addition and composition). ... Arturo Magidin ...
    (sci.math)
  • Re: Hom_R(R,R) question, rings
    ... > Tony wrote: ... > multiplication in Hom_Ris defined not by composition but by ... composition, Hom_R is a ring with 1." ... since rings don't require multiplicative inverses. ...
    (sci.math)