Security limitations of Shamir's secret sharing

The security is so important for both storing and transmitting the digital data, the choice of parameters is critical for a security system, that is, a weak parameter will make the scheme very vulnerable to attacks, for example the use of supersingular curves or anomalous curves leads to weaknesses in elliptic curve cryptosystems, for RSA cryptosystem there are some attacks for low public exponent or small private exponent. In certain circumstances the secret sharing scheme is required to decentralize the risk. In the context of the security of secret sharing schemes, it is known that for the scheme of Shamir, an unqualified set of shares cannot leak any information about the secret. This paper aims to show that the well-known Shamir's secret sharing is not always perfect and that the uniform randomization before sharing is insufficient to obtain a secure scheme. The second purpose of this paper is to give an explicit construction of weak polynomials for which the Shamir's (k, n) threshold scheme is insecure in the sense that there exist a fewer than k shares which can reconstruct the secret. Particular attention is given to the scheme whose threshold is less than or equal to 6. It also showed that for certain threshold k, the secret can be calculated by a pair of shares with the probability of 1/2. Finally, in order to address the mentioned vulnerabilities, several classes of polynomials should be avoided.