# Does zero-padding the secret in Shamir's sharing scheme increase security?

When performed in binary Galois fields $GF_{2^n}$, Shamir's threshold secret sharing scheme produces shares that are each the same bit-size as the secret. Though the scheme is "perfectly secure" in that less than the requisite number of shares provide no information about the secret, in practice it does leak information about the secret, namely its size (when performed in binary fields).

Would padding the binary representation of the secret (say to a bit-length that is a multiple of 1024, or something) increase the security of the scheme?

If random padding is better, when reconstructing the secret, what is the best way to know how many random bits were added? Should I have the first few bytes be a number that stores this info? Obviously, zero-padding does away with this problem, because any leading zeros can be dropped on reconstructing the secret.

I should add: I'm operating in $GF_{2^8}$, and computing each byte of the shares from a byte of the secret, with different random coefficients for each polynomial, as discussed in this question.

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