Question about modular arithmetic (division)

Question

I have a question about division operation.

There are two entities, Alice and Bob.

Bob randomly selects a finite set of numbers (set K). (about 10K or 100K)

Alice randomly generates one big number $x_1$ and chooses one element $y_1$ (from Bob's set K).

Alice makes $X$ ($X=x_1*y_1 \bmod n$) and sends $X$ to Bob.

In this case, can Bob figure out what Alice chooses from the set K?

Small number example: 3 ($x_1$) * 5 ($y_1$) mod 7 = 1 ($X$).

Bob knows that there is '5' in the set K, but I think that 1 divided by 5 (mod 7) equals 1.

So I don't think that Bob can figure out what Alice chooses from the set K.

Am I right?

Sorry for my bad English.

Thanks.

Answer 1

The answer depends on how $n$ is chosen.

If $n$ is coprime with each of the $y_j$ (including if $n$ is prime and all the $y_j$ are in range $[1..n[$), and $x_1\bmod n$ is uniformly random on $[0..n[$ (including if $x_1$ is uniformly random on that interval, or on a much larger interval), then demonstrably, Bob can not determine what $y_i$ Alice has picked by examination of $X\;=\;x_1*y_i\bmod n$. Proof sketch: the application $F_i:x_1\to X$ is a permutation of $[0..n[$; therefore $X$ is uniformly random on $[0..n[$, regardless of $y_i$; thus $X$ can't leak information about $y_i$.

However, if $n$ is the product of all the $y_i$ and these are distinct moderate primes, then Bob can find with fair odds which $y_i$ was chosen: $X$ is divisible by that $y_i$, and only those few (perhaps no) other $y_j$ that divide $X$.

There are gradations between theses two extremes.