[Cryptography] Mathematics of variable substitutions?

[Ron Garret]

On Apr 30, 2016, at 3:57 PM, Watson Ladd <watsonbladd at gmail.com> wrote:

> On Sat, Apr 30, 2016 at 4:17 AM, Bill Cox <waywardgeek at gmail.com> wrote:
>> I was hoping someone could point me in the direction of relevant mathematics
>> where we examine what equations can be converted to other equations using
>> variable substitutions, in ways that are efficiently computable modulo a
>> prime.  For example, we can easily convert an Edwards curve into a circle
>> with the substitution z^2 = x^2(1 + y^2).  However, this substitution does
>> not cause the Edwards addition law to become the circle group addition law.
>> It becomes something cool, but the equations are no more efficient than
>> computing the regular Edwards addition law.
>> 
>> Has it been proven that no birational substitution can convert the Edwards
>> addition law into the circle group addition law?  The circle group addition
>> law is:
> 
> See any book on algebraic geometry which proves the genus is a
> birational invariant.

Can you please elaborate on that a bit?  How does the non-existence of a birational substitution that converts Edwards to circles follow from the fact that the genus is a birational invariant?

rg