For the following cipher, what is the probability of someone without the private key generating a valid public key, using only information from a list of $k$ public keys previously generated with the private key?
This is the cipher:
To generate the private encryption key, $Y$: Let $X$ be an $n$ by $i$ matrix of random integers between $0$ and $9$, inclusive.
Let $Y$ be a vector of the $n$ real numbers defined by converting each row in $X$ to a real number between $0$ and $1$, e.g., $x_{1.} = (1, 2, 3)$ becomes $y_{1} = .123$.
To generate public decryption keys, $W$: Create a pair of random $j$-digit numbers between $0$ and $1$ inclusive, $a < b$. Let $Z =$ $R((Y - a/b)^2)$, where $R(.)$ returns the ascending rank order of reals, e.g., $R(23, 44, 2) = (2, 3, 1)$. Let $W = (a, b, Z)$.
To decrypt with public key: Test if $R((Y - w_{1}/w_{2})^2) = (w_{3}, w_{4}, ... , w_{n}).$
The probability of successfully generating a valid $W$ without any information about $Y$ is $1$ out of $n!$. What is the probability of successfully generating a valid $W$ with only the information from $k$ public keys previously generated from $Y$, in terms of $n$, $i$, $j$, and $k$?
Of note: According to @grand_chat's answer here, we can uniquely define any $Y$ as the sequence of solutions to the infinite series of functions $R((Y - r)^2)$, as $r$ ranges over the rational numbers from $min(Y)$ to $max(Y)$. This implies that one cannot deduce a unique $Y$ from any finite $k$ of distinct $W$, but also that the probability of generating a valid $W$ increases with increasing $k$.
[probability of correctly guessing W corrected from $1/10^n$ to $1/n!$ per response]
