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Threshold cryptography is the art of chopping a secret into little bits, so that the secret can only be learned by possessing more than a threshold number of those bits.
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Why is Shamir Secret Sharing not secure against active adversaries out-of-the-box?
In your example, let's assume the secret sharing scheme is a $(k,n)$-threshold sharing scheme with $k = \frac{n + 1}{2}$, as you say only an 'honest majority' can construct the secret.
If then $n$ pr …