Elliptic factoring with points *NOT* on the curve

The elliptic curve factoring algorithm seems to work with points
*NOT* on the curve.

Pick a random POINT=[x,y] that is not on the curve in Fp and perform point
addition and multiplication as if POINT were on the curve. The point
at infinity is hit, giving a cyclic group structure with order
different from the points on the curve.

As in this example:
<-->

p=19;
a=14;b=10;
e1=ellinit([0,0,0,a,b]);
x0=random(p);y0=random(p);while(ellisoncurve(e1,Mod(1,p)*[x0,y0]),x0=random(p);y0=random(p));
po1=Mod(1,p)*[x0,y0];print(" point=",po1," on curve=",ellisoncurve(e1,po1)," ellap=",ellap(e1,p));for(i=1,p+2,po2=centerlift(ellpow(e1,po1,i));print(i," ",po2));
<-->
point=[Mod(5, 19), Mod(2, 19)] on curve=0 ellap=3
1 [5, 2]
2 [-3, 5]
3 [2, 4]
4 [4, -9]
5 [-2, -1]
6 [3, 7]
7 [3, -7]
8 [-2, 1]
9 [4, 9]
10 [2, -4]
11 [-3, -5]
12 [5, -2]
13 [0]
14 [5, 2]
15 [-3, 5]
16 [2, 4]
17 [4, -9]
18 [-2, -1]
19 [3, 7]
20 [3, -7]
21 [-2, 1]

Points not on the curve with arbitrary even order seem to be
generated:

gp > po=ellpow(e1,[x,y],3);x0=random(p);tt=polrootsmod(subst(numerator(po[2]),x,x0),p);while(length(tt)<2,x0=random(p);tt=polrootsmod(subst(numerator(po[2]),x,x0),p));po2=[Mod(1,p)*x0,tt[2] ]
%244 = [Mod(8, 19), Mod(1, 19)]
gp > ellpow(e1,po2,2*3)
%245 = [0]
ellisoncurve(e1,po2)
%246 = 0
gp >

Some tests results that may be screwed by testing with small number:

1. Looks like all [x,y] not on the curve give group structure for
primes with good reduction.
2. There are points not on the curve with arbitrary even order.
3. Factoring not on the curve gives similar results with factoring on
the curve (at least to 10^20).

If this is right, what is known about the order of random [x,y] on a
random curve? (The classic algorithm has some restrictions)

Attached is modification to the elliptic curve factoring algorithm.

<-->

\\ This program is distributed under the terms of the GPL v2.
\\ The full text of the GPL is available at:
\\ http://www.gnu.org/licenses/gpl.txt

\\ Modification to the elliptic curve factoring algorithm working with points
\\ *NOT* on the curve

\\ Usage:
\\ gp
\\ \r FILENAME

{
facreal3(p,cu)=
local(e1,x0,y0,a,b,tt,i,po1,po2,o,c1,cp,l1,po);
a=random(p); b=random(p); po=x^3+a*x+b;
while(gcd(poldisc(po),p)!=1,a=random(p); b=random(p); po=x^3+a*x+b;);

e1=ellinit([0,0,0,a,b]);

for(c1=1,cu,
print(c1);
x0=random(p); y0=random(p);
while(ellisoncurve(e1,Mod(1,p)*[x0,y0]),
	x0=random(p); y0=random(p);
);

po1=Mod(1,p)*[x0,y0];
po2=po1;
print(" on curve=",ellisoncurve(e1,po1));
cp=0;
while(cp<B1,
cp=nextprime(cp+1);
p1=cp;
l1=round(log(B1)/log(p1)) ;
for(j=1,l1,
po2=(ellpow(e1,po2,p1));
if(centerlift(po2)==[0],o=i;print(" zero=",po2," cu=",c1));
);
);
);
return;
}

B1=11000;
p=1000000000000241*99999999999999990167;
facreal3(p,200);

<-->

[akruppa]

This is pushing my understanding of elliptic curves... but would a point not on E(p) be on the twist curve? Then it would be clear why you're getting group structure again.

Alex

Originally Posted by akruppa

This is pushing my understanding of elliptic curves... but would a point not on E(p) be on the twist curve? Then it would be clear why you're getting group structure again.

Alex

Are you sure about this?

In the example with:
p=19;
a=14;b=10;

There is a point of order 13.

The twisted curve has order 23 and the original curve has order 17.

Originally Posted by akruppa

but would a point not on E(p) be on the twist curve? Then it would be clear why you're getting group structure again.

Alex

If the coding is right: The pseudo point with constructed order of 6 [8,1] is not on any twist in F19

[akruppa]

I asked Pierrick Gaudry, who has worked on elliptic curve arithmetic before, about this and he replied simply that he didn't know about this idea, either. Someone may well have investigated this before, but if so, it's not very commonly known, apparantly. Sounds like an interesting topic to delve into some more.

Alex

[R.D. Silverman]

Originally Posted by akruppa

I asked Pierrick Gaudry, who has worked on elliptic curve arithmetic before, about this and he replied simply that he didn't know about this idea, either. Someone may well have investigated this before, but if so, it's not very commonly known, apparantly. Sounds like an interesting topic to delve into some more.

Alex

It is simple.

For a curve in the form y^2 = x^3 + ax + b, the parameter b does
not occur in the arithmetic when adding two points. So given
y^2 = x^3 + Ax + B for given A,B, a point (x1, y1) not on this curve
is instead on the curve y^2 = x^3 + Ax + B', where B' != B.

You are still adding points, but it is on a DIFFERENT curve from the one
you expect.