A Fuzzy Commitment Scheme

Alawi A. Al-saggaf Computer Science and Engineering College Al-Ahgaff University – Hadhramout Republic of Yemen alwiduh@yahoo.com Acharya H. S. Symbiosis Institute of Computer Studies and Research Symbiosis International University – Pune-India haridas.acharya@symbiosiscomputers.com

ABSTRACT This paper attempt has been made to explain a fuzzy commitment scheme. In the conventional Commitment schemes, both committed string m and valid opening key are required to enable the sender to prove the commitment. However there could be many instances where the transmission involves noise or minor errors arising purely because of the factors over which neither the sender nor the receiver have any control. The fuzzy commitment scheme presented in this paper is to accept the opening key that is close to the original one in suitable distance metric, but not necessarily identical. The concept itself is illustrated with the help of simple situation. KEY WORDS Cryptography, Error Correcting Codes, Fuzzy logic and Commitment scheme.

1. Introduction

The notion of Commitment scheme is at the heart of most the constructions of modern Cryptography protocols. Protocols are essentially a set of rules associated with a process or a scheme defining the process.
Commitment schemes are the processes in which the interests of the parties involved in a process are safeguarded and the process itself is made as fair as possible. Commitment protocols were first introduced by Blum [1] in 1982; many more Commitment Schemes were later developed with improved features [5, 6, 7, 8, 12, 13]. Moreover in the conventional Commitment schemes, opening key are required to enable the sender to prove the commitment. However there could be many instances where the transmission involves noise or minor errors arising purely because of the factors over which neither the sender nor the receiver have any control. Our aim in this paper to describe commitment schemes, which use algorithms to counter possible uncertainness. Uncertainty leads to introduction of fuzzy sets and fuzzy logic[2] in to the protocol itself. Fuzzy commitment scheme was first introduced by Juels and Martin [3], fuzziness also introduced later in [4,14,15] for generating cryptographic keys.

They add new property called “fuzziness” in the open phase to allow, acceptance of the commitment using corrupted opening key that is close to the original one in appropriate metric or distance. In this paper we have attempted a more formal and mathematical definition of fuzzy commitment schemes. An overview of commitment schemes and description of related work is also incorporated. A brief introduction of error correcting codes, with real life situation to illustrate is attempted.

2. Crisp Commitment Schemes In a conventional commitment scheme, one party, whom we denote the sender namely Alice, aim to entrust a concealed message m to the second party namely Bob. Intuitively a commitment scheme can be seen as the digital equivalent of a sealed envelope. If Alice wants to commit to some message m she just puts it into the sealed envelope, so that whenever Alice wants to reveal the message to Bob, she opens the envelope. Clearly, such a mechanism can be useful only if it meets some basic requirements. First of all the digital envelope should hide the message from: Bob should be able to learn m from the commitment (this is often referred in the literature as the hiding property). Second, the digital envelope should be binding, meaning with this that Alice can not change her mind about m, and by checking the opening of the commitment one can verify that the obtained value is actually the one Alice had in mind originally (this is often referred to as the binding property).
   Definition 1:A Commitment scheme is a tuple{P, E,M }
   Where M ={0,1}n is a message space, P is a set of individuals , generally with three elements A as the committing party, B as the party to which Commitment is made and TC as the trusted party ,
   E = { ( ti, ei) } are called the events occurring at times ti, i = 1,2,3 , as per algorithms ei , i = 1,2,3. The scheme always culminates in either acceptance or rejection by A and B. The environment is setup initially, according to the algorithm Setupalg (e1) and published to the parties A and B at time t1. During the Commit phase, A uses algorithm Commitalg (e2), which encapsulates a message mאM, along with secret string SאR{0,1}k into a string c. The opening key (secret key) could be formed using both m and S. A sends the result c to B( at tim

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