Can we share an n-bit string by using shamir's secret sharing n times? For Shamir's Secret Sharing, with a secret of size $n$, instead of having a finite field of order $n$, can we just use Shamir's Secret Sharing $n$ times, once for each bit, using a field of order 2?

Is this still as strong as using Shamir's Secret Sharing once with a field of order $n$? Are there any other drawbacks to this method?
For Shamir's Secret Sharing, with a secret of size $n$, instead of having a finite field of order $n$, can we just use Shamir's Secret Sharing $n$ times, once for each bit, using a field of order $2$?


If you are asking whether Shamir Secret Sharing works over the field $GF(2)$, well, no it doesn't (unless we have exactly one share); the problem is that, for each share, we need to give it a distinct nonzero $x$ coordinate.  For $GF(2)$, there's only one available value (1), and so, unless we're happy with there being only one share that we can distribute, it doesn't work.

However, if you mean 'can we use a field with a characteristic of 2' (say, $GF(2^8)$, yes, that works just great (and, in fact, what is commonly used in practice); $GF(2^8)$ gives us 255 possible shares, and that's often enough.

In addition, it is perfectly safe to split up a secret, and share each part separately; the only issue with your proposal is the size of the field.  Of course, if you were using $GF(2^8)$, you could split up the secret into 8 bit pieces, rather than a single bit, if you so choose.