Notation for commitment schemes

Question

I was reading this set of lecture notes on commitment schemes, where they define a commitment scheme $\text{Com}(b, r) = f(r), h(r) \oplus b$ as a secure commitment scheme. In this case, $f : \{0,1\}^n \rightarrow \{0,1\}^n$ is a one-way permutation, $h : \{0,1\}^n \rightarrow \{0,1\}$ is a hard-core bit of $f(\cdot)$, and $r$ appears to be a random string from $\{0,1\}^n$ (the notation is $r \leftarrow_R \{0,1\}^n$, but I'm not really sure what this actually means and it would be super helpful if someone could clarify what the notation means here!).

My question is what the $f(r), h(r) \oplus b$ means. What is the comma doing? Does this mean that the commitment scheme returns two outputs (I was under the impression that commitment schemes only return one output)? Or am I supposed to combine $f(r)$ with $h(r) \oplus b$ somehow? Thank you for your help!

Answer 1

Or am I supposed to combine $f(r)$ with $h(r) \oplus b$ somehow?

Yes, $\text{Com}(b, r)$ is a function which "combines" them. However it only "combines" them in the sense that it outputs the pair $f(r), h(r) \oplus b$

The main confusion here stems from not realizing that a tuple can be a perfectly valid singular output. (This is compounded by Boaz Barak writing the tuple without parentheses.)

From the lecture notes:

Definition 2 (Commitment schemes). A commitment scheme $Com$ is an unkeyed function that takes two inputs: a plaintext $x ∈ \{0,1\}^n$ ` and randomness $r$ chosen in $\{0,1\}^n$. The idea is that to commit to the winner I let $x$ be my prediction, choose $r \leftarrow_R \{0,1\}^n$ n and publish $y = Com(x, r)$. Later to prove I predicted $x$, I will publish $x$ and $r$.

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(the notation is $r \leftarrow_R \{0,1\}^n$, but I'm not really sure what this actually means and it would be super helpful if someone could clarify what the notation means here!).

On page 3 of the notes, the author uses $R$ to indicate random. $\gets$ is assignment. $\gets_R$ is used for uniform random assignment.

For more see this answer.