[attachment=12595]
Abstract:-
Authenticity is one of the key aspects of computer security. As a cryptographic technique that guarantees information authenticity, secret sharing has been an active research field for many years. With the efforts of pioneering researchers, secret sharing has reached the level of practical use in today's Internet-based applications, which is designed to protect a secret piece of information among a group of users in such a way that only certain subsets of users can jointly reconstruct the secret, whereas other subsets of users can ideally not obtain any information about the secret.
In this paper , we present an online secret sharing schemes and investigate the design of online secret sharing schemes . A new model is then proposed for computationally secure online secret sharing, which allow to change dynamically the secret and add participants, without having to redistribute new shares secretly to the current participants.The security of the scheme is based on the intractability of factoring. This paper include the advantages provided by online secret sharing.We have also discussed about how cheating can occur and how to identify all the cheaters.
1 . Introduction:-
As long as there are creatures endowed with language, there will be confidential messages intended for a limited audience. How can these messages be transmitted secretly, so that no unauthorised person gets knowledge of the content of message? And how can one guarantee that a message arrives in the right hands exactly as it was transmitted? For this purpose we use the different cryptography techniques . Traditionally, there are two ways to answer such questions. One can disguise the very existence of a message, perhaps by writing with invisible ink; or try to transmit the message via a trustworthy person. A totally different approach is to encipher (or encrypt) a message. In this case one does not disguise its existence. There is a satisfying appropriateness to cryptology’s role in the birth of electronic computing. The arrival of the Information Age has revealed an urgent need for cryptography in the private sector. Today, vast amounts of sensitive information such as health and legal records, financial transactions, credit ratings and the like are routinely exchanged between computers via public communication facilities. Society turns to the cryptographer for help in ensuring the privacy and authenticity of such sensitive information.
Cryptographic techniques, such as encipherment, digital signatures, key agreement and secret sharing schemes, are important building blocks in the implementation of any security service. A secret sharing scheme is a system that is designed to protect a secret piece of information among a group of users in such a way that only certain subsets of users can jointly reconstruct the secret, whereas other subsets of users can ideally not obtain any information about the secret. The collection of subsets that can access the secret is called the access structure of the secret sharing scheme. The most common types of secret sharing schemes are those where the access structure consists of all subsets of at least k (out of n) users (these are known as (k,n)- threshold schemes).
2. Secret sharing:-
Secret sharing is an important and widely studied tool in cryptography and distributed computation. Informally, a secret sharing scheme is a protocol in which a dealer distributes a secret among a set of participants such that only specific subsets of them, defined by the access structure, can recover the secret at a later time. Much research in the area of secret sharing has concentrated on the size of the shares. Although the size of the shares is important because the shares have to be transmitted and stored secretly, this is not the only information the participants must know to reconstruct the secret. Additional knowledge needed includes, for example, the identity of the participants and the description of the protocol, including the access structure. These parameters are publicly known, but at the same time it is vital that they are authentic, i.e. that no malicious participant has changed these descriptions.
The original motivation for secret sharing is the following.
1. To safeguard cryptographic keys from loss, it is desirable to create backup copies, although these copies are themselves a security risk. Secret sharing addresses this issue by allowing enhanced reliability without increased risk.
2. They also facilitate distributed trust or shared control for critical activities by requiring cooperation by t out of n users for access to a critical action.
The idea of secret sharing is to start with a secret, and divide it into pieces called shares which are distributed amongst users such that the pooled shares of specific subsets of users allow reconstruction of the original secret. This may be viewed as a key distribution technique, facilitating one-time key establishment, wherein the recovered key is pre-determined (static), and in the basic case, the same for all groups.
2.1 Definitions Of Secret Sharing:-
1. A secret sharing scheme is a protocol involving a set A ={A1,……., An} of participants and a dealer D, where
D  P. Let T  2P be the set of subsets of participants permitted access to the secret; this is called the access structure. The dealer D chooses a secret K and distributes privately to each participant Ai Є P a share Si of K such that:
(i) any authorized set X Є T can reconstruct the secret K from its shares,
(ii) no unauthorized set X  T can do so.
2. Let T*  T be the set of minimal authorised sets, that is, of sets X Є T such that: Y  X and Y ЄT implies that Y = X.
3. A secret sharing scheme is perfect if the shares corresponding to each unauthorized subset provide absolutely no information about the shared secret.
2.2 The Shamir threshold scheme:-
The Shamir (t; n)-threshold scheme is based on polynomial interpolation over a finite field Fp, where p is a prime, and the fact that a polynomial f(x) of degree t ─ 1 is uniquely determined by a set of t pairs (xi; f(xi)), where all xi are distinct. Let the set of possible secrets be P = Zp, where p ¸ t + 1 is prime, and let S = Zp. The secret key will be an element of Zp, as will be the shares given to the participants. The Shamir threshold scheme is described as follows:
(i) Initialization phase
D chooses t distinct, non zero elements of Zp, denoted xi, 1 ≤ i ≤ t (this is where we require p ≥ t + 1).
For 1 ≤ i ≤ t, D gives the value xi to Pi. The values xi are public.
(ii) Distribution phase
1. Suppose D wants to share a key K Є Zp. D secretly chooses (independently at random) t ─ 1 elements of Zp, a,…….., at ─ 1.
2. For 1 ≤ i ≤ t, D computes yi = f(xi), where
3. For 1 ≤ i ≤ t, D gives the share yi to Pi.
(iii) Reconstruction phase
Any set X 2 ¡ of t where X = {xi1 ,………., xit} or more participants can reconstruct the secret key K = f(0) by substituting x = 0 into the Lagrange interpolation formula:
2.3 Secret sharing schemes with extended capabilities:-
Secret sharing schemes with a variety of extended capabilities exist, including:
(i) pre-position secret sharing schemes. All necessary secret information is put in place excepting a single (constant) share which must later be communicated, e.g., by broadcast, to activate the scheme.
(ii) dynamic secret sharing schemes. There are pre-positioned schemes wherein the secrets reconstructed by various authorised subsets vary with the value of communicated activating shares.
(iii) multi-secret threshold schemes. In these secret sharing schemes different secrets are associated with different authorised subsets.
(iv) detection of cheaters and verifiable secret sharing. These schemes respectively address cheating by one or more group members, and the distributor of the shares.
(v) secret sharing with disenrollment. These schemes address the issue that when a secret share of a (t; n)threshold scheme is made public, it becomes a (t ─ 1; n) scheme.
4 Online Secret Sharing Schemes:-
Online secret sharing schemes provide the capability to dynamically change the secret and add participants, without having to redistribute new shares secretly to the current participants. These capabilities are traded for the need to store online additional authentic (but not secret) information at a publicly accessible location, e.g. on a notice board. Alternatively, this information can be broadcast to the participants over a public channel. In particular, online secret sharing schemes have the following properties:
(i) All shares that must be transmitted and stored secretly once for every participant are as short as the secret.
(ii) Different secrets can be shared with different access structures without requiring the shares held by participants to change. This includes the ability for the dealer to change the secret after the shares have been distributed.
(iii) The dealer can update the scheme online: When a new participant is added and the access structure is changed, already distributed shares remain valid.