Secret sharing

For cases when the whole secret is known by all participants, see shared secret.

Secret sharing refers to method for distributing a secret amongst a group of participants, each of whom is allocated a share of the secret. The secret can be reconstructed only when a sufficient number of shares are combined together; individual shares are of no use on their own.

More formally, in a secret sharing scheme there is one dealer and n players. The dealer gives a secret to the players, but only when specific conditions are fulfilled. 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 by both Adi Shamir and George Blakley, independent of each other, in 1979.

Contents

Importance of Secret Sharing Schemes

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 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 scheme

A secure secret sharing scheme distributes shares so that anyone with fewer than t shares has no extra information about the secret than someone with 0 shares. Consider the naive 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. He 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. This system is not a secure secret sharing scheme, because a player with fewer than t shares gains significant information about the content of the secret. In a secure scheme, even a player missing only one share should still face 268 = 208 billion combinations.

Limitations of secret sharing schemes

Several secret sharing schemes are said to be information theoretically secure and can be proved 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 128-bits of entropy 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:

Trivial secret sharing

There are several (t, n) secret sharing schemes for t = n, when all shares are necessary to recover the secret:

When space efficiency is not a concern, these schemes can be used to reveal a secret to any desired subsets of the players simply by applying the scheme for each subset. For example, to reveal a secret s to any two of the three players Alice, Bob and Carol, create three different (2,2) secret shares for s, giving the three sets of two shares to Alice and Bob, Alice and Carol, and Bob and Carol. This approach quickly becomes impractical as the number of subsets increases, for example when revealing a secret to any 50 of 100 players, whereas the schemes described below allow secrets to efficiently be shared with a threshold of players.

A t ≠ n example

The difficulty lies in creating schemes that are still secure, but do not require all n shares. For example, imagine that the Board of Directors of a company would like to protect their secret formula. The president of the company should be able to access the formula when needed, but in an emergency any 3 of the 12 board members would be able to unlock the secret formula together. This can be accomplished by a secret sharing scheme with t = 3 and n = 15, where 3 shares are given to the president, and 1 is given to each board member.

Shamir's scheme

Main article: Shamir's Secret Sharing

In this scheme, any t out of n shares may be used to recover the secret. The system relies on the idea that you can fit a unique polynomial of degree (t-1) to any set of t points that lie on the polynomial. It takes two points to define a straight line, three points to fully define a quadratic, four points to define a cubic curve, and so on. That is it takes t points to define a polynomial of degree t-1. The method is to create a polynomial of degree t-1 with the secret as the first coefficient and the remaining coefficients picked at random. Next find n points on the curve and give one to each of the players. When at least t out of the n players reveal their points, there is sufficient information to fit a (t-1)th degree polynomial to them, the first coefficient being the secret.

Blakley's scheme

Two nonparallel lines in the same plane intersect at exactly one point. Three "nonparallel" planes in space intersect at exactly one point. More generally, any n nonparallel n-dimensional hyperplanes intersect at a specific point. The secret may be encoded as any single coordinate of the point of intersection. If the secret is encoded using all the coordinates, even if they are random, then an insider (someone in possession of one or more of the n-dimensional hyperplanes) gains information about the secret since he knows it must lie on his plane. If an insider can gain any more knowledge about the secret than an outsider can, then the system no longer has information theoretic security. If only one of the n coordinates is used, then the insider knows no more than an outsider (i.e., that the secret must lie on the x-axis for a 2-dimensional system). Each player is given enough information to define a hyperplane; the secret is recovered by calculating the planes' point of intersection and then taking a specified coordinate of that intersection.

Blakley's scheme in three dimensions: each share is a plane, and the secret is the point at which three shares intersect. Two shares are insufficient to determine the secret, although they do provide enough information to narrow it down to the line where both planes intersect.

Blakley's scheme is less space-efficient than Shamir's; while Shamir's shares are each only as large as the original secret, Blakley's shares are t times larger, where t is the threshold number of players. Blakley's scheme can be tightened by adding restrictions on which planes are usable as shares. The resulting scheme is equivalent to Shamir's polynomial system.

Secret Sharing using the Chinese Remainder Theorem

Main article: Secret Sharing using the Chinese Remainder Theorem

The Chinese Remainder Theorem can also be used in secret sharing, for it provides us with a method to uniquely determine a number S modulo k many relatively prime integers m_1, m_2, ..., m_k, given that S < \prod_{i=1}^k m_i. There are two secret sharing schemes that make use of the Chinese Remainder Theorem, Mignotte's and Asmuth-Bloom's Schemes. They are threshold secret sharing schemes, in which the shares are generated by reduction modulo the integers m_i, and the secret is recovered by essentially solving the system of congruences using the Chinese Remainder Theorem.

Proactive secret sharing

Main article: Proactive secret sharing

If the players store their shares on insecure computer servers, an attacker could crack in and steal the shares. If it is not practical to change the secret, the uncompromised (Shamir-style) shares can be renewed. The dealer generates a new random polynomial with constant term zero and calculates for each remaining player a new ordered pair, where the x-coordinates of the old and new pairs are the same. Each player then adds the old and new y-coordinates to each other and keeps the result as the new y-coordinate of the secret.

All of the non-updated shares the attacker accumulated become useless. An attacker can only recover the secret if he can find enough other non-updated shares to reach the threshold. This situation should not happen because the players deleted their old shares. Additionally, an attacker cannot recover any information about the original secret from the update files because they contain only random information.

The dealer can change the threshold number while distributing updates, but must always remain vigilant of players keeping expired shares.

Verifiable secret sharing

Main article: Verifiable secret sharing

A player might lie about his own share to gain access to other shares. A verifiable secret sharing (VSS) scheme allows players to be certain that no other players are lying about the contents of their shares, up to a reasonable probability of error. Such schemes cannot be computed conventionally; the players must collectively add and multiply numbers without any individual's knowing what exactly is being added and multiplied. Tal Rabin and Michael Ben-Or devised a multiparty computing (MPC) system that allows players to detect dishonesty on the part of the dealer or on part of up to one third of the threshold number of players, even if those players are coordinated by an "adaptive" attacker who can change strategies in realtime depending on what information has been revealed.

Computationally Secure Secret Sharing

The disadvantage of unconditionally secure secret sharing schemes is that the storage and transmission of the shares requires an amount of storage and bandwidth resources equivalent to the size of the secret times the number of shares. If the size of the secret were significant, say 1 GB, and the number of shares were 10, then 10 GB of data must be stored by the shareholders. Alternate techniques have been proposed for greatly increasing the efficiency of secret sharing schemes, by giving up the requirement of unconditional security.

One of these techniques, known as secret sharing made short,[1] combines Rabin's information dispersal algorithm[2] (IDA) with Shamir's secret sharing. Data is first encrypted with a randomly generated key, using a symmetric encryption algorithm. Next this data is split into N pieces using Rabin's IDA. This IDA is configured with a threshold, in a manner similar to secret sharing schemes, but unlike secret sharing schemes the size of the resulting data grows by a factor of (number of fragments / threshold). For example, if the threshold were 10, and the number of IDA-produced fragments were 15, the total size of all the fragments would be (15/10) or 1.5 times the size of the original input. In this case, this scheme is 10 times more efficient than if Shamir's scheme had been applied directly on the data. The final step in secret sharing made short is to use Shamir secret sharing to produce shares of the randomly generated symmetric key (which is typically on the order of 16 - 32 bytes) and then give one share and one fragment to each shareholder.

A related approach, known as AONT-RS,[3] applies an All-or-nothing transform to the data as a pre-processing step to an IDA. The All-or-nothing transform guarantees that any number of shares less than the threshold is insufficient to decrypt the data.

Other uses and applications

A secret sharing scheme can secure a secret over multiple servers and remain recoverable despite multiple server failures. The dealer may treat himself as several distinct participants, distributing the shares between himself. Each share may be stored on a different server, but the dealer can recover the secret even if several servers break down as long as he can recover at least t shares; however, crackers that break into one server would still not know the secret as long as less than t shares are stored on each server.

This is one of the major concepts behind the Vanish computer project at the University of Washington, where a random key is used to encrypt data, and the key is distributed as a secret across several nodes in a P2P network. In order to decrypt the message, at least t nodes on the network must be accessible; the principle for this particular project being that the number of secret-sharing nodes on the network will decrease naturally over time, therefore causing the secret to eventually vanish. However, the network is vulnerable to a Sybil attack, thus making Vanish insecure.[4]

A dealer could send t shares, all of which are necessary to recover the original secret, to a single recipient. An attacker would have to intercept all t shares to recover the secret, a task which is more difficult than intercepting a single file, especially if the shares are sent using different media (e.g. some over the Internet, some mailed on CDs).

For large secrets, it may be more efficient to encrypt the secret and then distribute the key using secret sharing.

Secret sharing is an important primitive in several protocols for secure multiparty computation.

See also

References

  1. ^ Krawczyk, Hugo (1993). "Secret Sharing Made Short". CRYPTO '93. http://www.cs.cornell.edu/courses/cs754/2001fa/secretshort.pdf. 
  2. ^ Rabin, Michael O. (1989). "Efficient dispersal of information for security, load balancing, and fault tolerance". Journal of the ACM 36 (2). 
  3. ^ Resch, Jason; Plank, James (February 15, 2011). "AONT-RS: Blending Security and Performance in Dispersed Storage Systems". Usenix FAST'11. http://web.eecs.utk.edu/~plank/plank/papers/FAST-2011.pdf. 
  4. ^ "'Unvanish: Reconstructing Self-Destructing Data'". http://z.cs.utexas.edu/users/osa/unvanish/home. 

External links