Polynomial for secure communication Can we use a polynomial $P(x)$ to securely communicate ?

For example Alice &amp; Bob agree on a degree of a polynomial. The value $x$ &amp; the coeficients are generated from a seed using a hash function or any other PRG and change in each communication, the plaintext $m$ is hidden into $P(x)$. For example  $P(x) = ax^2+b(m)x+c$

alice sends the result of $P(x)$ to Bob, once received bob retreive $m$ from $P(x)$

can we provide a certain security level with the above method ?
If you're operating in some finite field, and the PRG used to generate the coefficients and $x$ is cryptographically secure (that is, it is infeasible to distinguish the output of the PRG from a truly random source of field elements), and you go back to the PRG for every single message, then it is secure as far as privacy is concerned (of course, you need to make sure that $bx$ is nonzero; otherwise, it might be a bit tricky for the decryptor to recover $m$).

As for integrity, well, if the attacker replaces a ciphertext $C$ with another $C'$, then that will decrypt to some message $m'$, where $m - m' = I(C - C')$ for some invertible element $I$ which is unknown to the attacker.  This is why you want to do the operation in a finite field (rather than a finite ring); in a field, $I$ can be any nonzero value, and so while the attacker can make the altered ciphertext decrypt to something else other than m, but he literally has no control beyond that.

On the other hand, the simpler polynomial $P(x) = mx + a$ has the same property; if $a$ and $m$ is generated via cryptographically secure source and you're operating in a finite field, then transmitting $mx+a$ also provides privacy, and prevents the attacker from making nonrandom changes to the decrypted ciphertext.

This garbling property (where any change in the ciphertext will make a random change in the plaintext) is the best we can do if the ciphertext must be the same size as the plaintext.  Generally, that is not considered good enough, and usually we include an authenticate operation which makes the ciphertext strictly larger than the plaintext, but also makes sure that any change in the ciphertext will cause the decryptor to reject the modified message (rather than emitting a random decryption).