I thought about this idea as a method of cryptography. I appreciate if someone could advice if it is wrong.
The method applies a SVD (singular value decomposition) method known in the linear algebra.
It still applies the trap-door algorithm in such way that Alice should send the public key to Bob but keeps the private key for herself. Later she will use the private key to open the message received from Bob.
Here is the sketch of the method:
1) Alice constructs a matrix $M$; consists of $m$ rows and $n$ columns, where $m$ and $n$ are large numbers and very close in the number of digits. Say (1000 x 995). The matrix entries are complex numbers.
2) Alice uses SVD to decompose $M$ yields $M=USV^T$, where $S$ is the matrix with complex eigen values.
3) Alice shares the public key which is $U^TS^{-1}$ where $U^T$ is the transpose of $U$ and $S^{-1}$ is the inverse of $S$.
4) Bob constructs his message in the form of a column vector $Q$ with entries of natural numbers, and he performs $[QU^TS^{-1}]^T$ and gets $q^T$. Note that $q$ is reduced vector on a lower dimension space with length, m or n whichever shorter in number of digits. Bob then sends $q$ to Alice.
5) Alice retrieves $Q`$ by using; $Q`=USq$.
Because $m$ and $n$ are large but unequal in the numbers of digits, $U^TS^{-1}$ can not be a square matrix. This will not allow Eve, a spy agent, to invert it and get $US$.
Eve also, can not retrieve neither $U$ nor $S$ by multiplying $U^TS^{-1}$ by its transpose. This because when doing so, she may only get the value modulus of the complex number representing the eigen values of $S$ but she can not get argument (the phase factor).
And because $m$ and $n$ are so large, the omitting of the smaller $m-n$ number of the eigen values will not affect the round approximation $Q`$ to $Q$.