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In step 3 of Simon's algorithm, we are told to "Repeat until there are enough such $y$’s that we can classically solve for $s$."

It then goes on:

enter image description here

The above are from this course notes.

I am not sure how this probability was calculated. Especially, why are there $2^{n-1}$ $y$'s such that $$y \cdot s = y_1s_1+y_2s_2 +\cdots y_ns_n = 0$$

I understand that there are n-1 non-trivial, linearly independent solutions to $y \cdot s = 0$, but how is $2^{n-1}$ obtained?

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    $\begingroup$ it would be better if you could spell out (in the post, not in the comments) your current understanding of the algorithm. This helps people know where exactly your misunderstanding lies $\endgroup$
    – glS
    Jun 23 at 8:24
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    $\begingroup$ And can you link to the paper where you found that probability so people can look at how it's derived? $\endgroup$
    – epelaaez
    Jun 23 at 15:40
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It is a general fact from linear algebra that for a non-zero vector $v$ in an $n$-dimensional vector space $V$ the subset

$$ A_v = \{u\in V \,|\, \langle u, v\rangle = 0\} $$

is an $(n-1)$-dimensional subspace of $V$. The fact can be proven easily by extending $\{v\}$ to an orthonormal basis.

Thus, in the specific case of the dot product $y \cdot s$ the subset $A_s = \{y \,|\, y \cdot s = 0\}$ is the $(n-1)$-dimensional vector space over $\mathbb{F}_2$.

Now, any two vector spaces of the same finite dimension over the same scalar field are isomorphic. Therefore, $A_s$ is isomorphic to $\mathbb{F}^{n-1}_2$ which consists of all binary sequences of length $n-1$. The isomorphism is a bijection, so $|A_s| = |\mathbb{F}^{n-1}_2|$. Conclusion follows from the fact that $|\mathbb{F}^{n-1}_2|=2^{n-1}$.

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