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The notion of imperfect (ramp) secret sharing allows one to share multiple secrets in one go with a lower overhead in shares than threshold secret sharing. I believe these might answer your question: Ramp Cheater Identification here Ramp Cheating Detection here, here and here

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you have several plaintext texts and their encryption, you also have the public key, is it possible some form of attack to find the private key? It depends in particular on how the plaintexts/ciphertexts pairs are chosen. As an extreme example, if a plaintext is the private exponent $d$ (which the statement does not forbid), then yes it's possible to find ...

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But the question is, can the Chinese remainder theorem in ECDSA be applied to the parameters in secp256k1? That precise attack doesn't work - we don't use the Chinese remainder theorem when computing with secp256k1 (as the group order is prime). On the other hand, there are certainly side channel attacks available against naïve implementations of ECDSA and ...

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The Coppersmith method, as usually stated with the $\epsilon$ factor, finds a root of a polynomial $f(x)$ of degree $d$ modulo $n$ of size $x \le n^{\frac{1}{d} - \epsilon}$, $0 < \epsilon \le 1/7$. The Håstad attack with $e = 11$ is fundamentally an application of the Coppersmith method with $f(x)$ of degree $11$ modulo $n_0\cdot n_1 \cdot \dots n_{e-1}$....

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