Secret sharing (also called secret splitting) refers to methods for distributing a secret among a group, in such a way that no individual holds any intelligible information about the secret, but when a sufficient number of individuals combine their 'shares', the secret may be reconstructed. Whereas insecure secret sharing allows an attacker to gain more information with each share, secure secret sharing is 'all or nothing' (where 'all' means the necessary number of shares). In one type of secret sharing scheme there is one dealer and n players. The dealer gives a share of the secret to the players, but only when specific conditions are fulfilled will the players be able to reconstruct the secret from their shares. The dealer accomplishes this by giving each player a share in such a way that any group of t (for threshold) or more players can together reconstruct the secret but no group of fewer than t players can. Such a system is called a (t, n)-threshold scheme (sometimes it is written as an (n, t)-threshold scheme). Secret sharing was invented independently by Adi Shamir and George Blakley in 1979.
Importance Secret sharing schemes are ideal for storing information that is highly sensitive and highly important. Examples include: encryption keys, missile launch codes, and numbered bank accounts. Each of these pieces of information must be kept highly confidential, as their exposure could be disastrous; however, it is also critical that they should not be lost. Traditional methods for encryption are ill-suited for simultaneously achieving high levels of confidentiality and reliability. This is because when storing the encryption key, one must choose between keeping a single copy of the key in one location for maximum secrecy, or keeping multiple copies of the key in different locations for greater reliability. Increasing reliability of the key by storing multiple copies lowers confidentiality by creating additional attack vectors; there are more opportunities for a copy to fall into the wrong hands. Secret sharing schemes address this problem, and allow arbitrarily high levels of confidentiality and reliability to be achieved. Secret sharing also allows the distributor of the secret to trust a group 'in aggregate'. Traditionally, giving a secret to a group for safekeeping would require that the distributor completely trust all members of the group. Secret sharing schemes allow the distributor to securely store the secret with the group even if not all members can be trusted all the time. So long as the number of traitors is never more than the critical number needed to reconstruct the secret, the secret is safe. Secret sharing schemes are important in cloud computing environments. Thus a key can be distributed over many servers by a threshold secret sharing mechanism. The key is then reconstructed when needed. Secret sharing has also been suggested for sensor networks where the links are liable to be tapped, by sending the data in shares which makes the task of the eavesdropper harder. The security in such environments can be made greater by continuous changing of the way the shares are constructed.
"Secure" versus "insecure" secret sharing A secure secret sharing scheme distributes shares so that anyone with fewer than t shares has no more information about the secret than someone with 0 shares. Consider for example the secret sharing scheme in which the secret phrase "password" is divided into the shares "pa––––––", "––ss––––", "––––wo––", and "––––––rd". A person with 0 shares knows only that the password consists of eight letters, and thus would have to guess the password from 268 = 208 billion possible combinations. A person with one share, however, would have to guess only the six letters, from 266 = 308 million combinations, and so on as more persons collude. Consequently, this system is not a "secure" secret sharing scheme, because a player with fewer than t secret shares is able to reduce the problem of obtaining the inner secret without first needing to obtain all of the necessary shares. In contrast, consider the secret sharing scheme where X is the secret to be shared, Pi are public asymmetric encryption keys and Qi their corresponding private keys. Each player J is provided with {P1(P2(...(PN(X)))), Qj}. In this scheme, any player with private key 1 can remove the outer layer of encryption, a player with keys 1 and 2 can remove the first and second layer, and so on. A player with fewer than N keys can never fully reach the secret X without first needing to decrypt a public-key-encrypted blob for which he does not have the corresponding private key – a problem that is currently believed to be computationally infeasible. Additionally we can see that any user with all N private keys is able to decrypt all of the outer layers to obtain X, the secret, and consequently this system is a secure secret distribution system.
Limitations Several secret-sharing schemes are said to be information-theoretically secure and can be proven to be so, while others give up this unconditional security for improved efficiency while maintaining enough security to be considered as secure as other common cryptographic primitives. For example, they might allow secrets to be protected by shares with entropy of 128 bits each, since each share would be considered enough to stymie any conceivable present-day adversary, requiring a brute force attack of average size 2127. Common to all unconditionally secure secret sharing schemes, there are limitations:
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