2*2 Matrix with determinant 1

[bhelmes]

A peaceful night for you,

do I see it right that 2*2 matrix with determinant 1 build a subgroup in linear algebra
and that calculation in the group is better for factoring than calculation in all 2*2 matrix ?

Do I see right that the matrix
(2 0)
(0 -1) has the determinant -2 and that an exponentation with 2*p of Mp mod Mp will result in a matrix with determinant 1

Is it rigth that there are 3 pauli-matrixs
https://en.wikipedia.org/wiki/Pauli_matrices
with determinant 1 and that a parallel multiplication of a start matrix with determinant 1 and a pauli matrix will speed up a possible factorisation ?

Would be nice to get a reply.

P.S. I do not know why or how, but I am sure we will find the next Mp before next christmas.

[Nick]

Over any commutative ring R, the 2x2 matrices with determinant 1 form a subgroup \(SL_2(R)\)
of the unit group of the ring of all 2x2 matrices.
The Pauli matrices have determinant -1, however.
It is important to decide carefully which commutative ring R you are working over:
the integers, the Gaussian integers, the complex numbers, etc.
Some of these can be constructed using matrices: for example, matrices of the
form
\[ \left(\begin{array}{rr}x & -y \\ y & x\end{array}\right)\]
over the real numbers form a copy of the complex numbers.

[Dr Sardonicus]

Let's see. det(M^k) = (det(M))^k.

So if det(M) = -2, det(M^2p) = (-2)^2p = 2^2p == 1 (mod 2^p - 1). Check.

I don't see any relation to factorization.

[bhelmes]

Originally Posted by Nick

Over any commutative ring R, the 2x2 matrices with determinant 1 form a subgroup \(SL_2(R)\)
of the unit group of the ring of all 2x2 matrices.

Considering the matrix M of the form
(n²+1; n)
(n; 1)

=(a; b)
(c; d)

The determinant is one.

For n=0 it includes the neutral element concerning the multiplication,

is closed under exponentation,
means for M^n the elements of b and c are equal and

Is closed under multiplication for the element a=1 mod 4.
It has obviously an inverse matrix
and the multiplication of M*N seems to be kommutativ.

It is not clear for me how you calculate the group order of this sub-sub-group
if you limit all elements modulo f and limit the multiplication to the exponentation of one M

Perhaps someone could correct me and add a little bit more knowledge.

Thanks in advance.






P.S. I managed to speed up my computer ; 7 days instead 8 days for an exponent at the wawefront.
(137 Watt instead 120 Watt) (you could easily speed up the calculation in the bios (Asus mainboard) )

[Dr Sardonicus]

Originally Posted by bhelmes

<snip>
It is not clear for me how you calculate the group order of this sub-sub-group
if you limit all elements modulo f and limit the multiplication to the exponentation of one M

The characteristic polynomial of M = [n^2 + 1, n; n, 1] is F(x) = x^2 - (n^2 + 2)*x + 1. The discriminant is D = n^2*(n^2 + 4).

Since M is a real symmetric matrix, it is diagonalizable, with diagonal elements the roots of F(x) = 0.

If f = p, a prime number, and gcd(p,D) = 1, then the multiplicative order of M (mod p) divides p - 1 if (D/p) = +1 (quadratic character).

Because F(x) is a quadratic with constant term 1, we can say that the multiplicative order divides p + 1 if (D/p) = -1.

So we can say that if p does not divide n or n^2 + 4, the multiplicative order (mod p) divides p - (D/p).

EDIT: Assuming p does not divide D, we can drop the square factor n^2 from the quadratic residue symbol, and use instead of (D/p).