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Adaptive Universal Composability Framework for Server-Aided Threshold Signature

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Adaptive Universal Composability Framework for Server-Aided Threshold Signature Research Journal of Applied Sciences, Engineering and Technology 5(1): 270-274, 2013 ISSN: 2040-7459; e-ISSN: 2040-7467 © Maxwell Scientific Organization, 2013 Submitted: June 02, 2012 Accepted: June 23, 2012 Published: January 01, 2013 Adaptive Universal Composability Framework for Server-Aided Threshold Signature Xuan Hong and Luqun Li Department of Computer Science and Technology, Shanghai Normal University, Shanghai 200234, China Abstract: The threshold signature scheme is a protocol that allows any subset of t parties out of n to generate a signature. Since the t members can cooperate together to compute the secret key, we introduce the server-aided threshold signature, which provides controllability for activating the signing function in a certain enhanced way. In this study, we present a server-aided threshold RSA signature protocol against the adaptive attacks. We give the universally composable secure model for the server-aided threshold signature primitive and prove that the proposed protocol is claimed to be well-formed, correct and unforgeability. As a separate contribution we also prove that it is also secure in the adaptive universal composability framework. After the discussion about the security and the performance, we claim that the protocol is practical and efficient. Keywords: Adaptive security, server-aided, threshold signature, universal composability framework INTRODUCTION that the password is divided into several pieces which are shared among a group of members. Only at least t A threshold signature scheme is a protocol that members together have the privilege to access the allows any subset of t parties out of n to generate a classified file. However, when some t members are signature. Meanwhile, it disallows the creation of a corrupted, the file will be leaked out. valid signature if fewer than t parties participate in the Therefore, we introduce the verifiable server, protocol. It should also be robust, meaning that which is physical secure and can be trusted by the corrupted parties should not be able to prevent company gateway. The server-aided threshold signature uncorrupted parties from generating signature. Though scheme, where the signing function can be activated by threshold schemes based on the discrete logarithm a server, provides user controllability for activating the problem are relatively straightforward to build, Basing user’s signing function. That is, if anyone has a power threshold schemes on the RSA problem is more to activate the signing function of the servers, he can difficult, due to the fact that the modulus Ф (N) cannot easily compute valid signatures for a specific be leaked to any of the shareholders. Boyd (1989) and organization without knowing the private key. Xu and Frankel (1989) present the first RSA based threshold Sandhu (2003) provided a general construction to build signature independently. These earlier protocols server assisted threshold signatures. Yang et al. (2006) additively share the signing key d among the parties. Rabin (1998) present the first reasonable efficient and also design two types of proactive secret sharing robust solutions. However, they tend to be more schemes which are suitable for server assisted threshold complex and less efficient in comparison to the discrete signatures. logarithm schemes. These earlier protocols additively In this study, we propose a server-aided threshold share the signing key d among the parties and they tend RSA signature protocol with adaptive universally to be more complex and less efficient. Later, (Shoup, composable security. In this protocol, the system is 2000) described his practical threshold signatures, composed of one server and multiple users who share which is widely regarded as the efficient threshold RSA the private key together. The server and all users must signature scheme. cooperate to produce a valid signature. In addition, the We consider a scenario that a classified file is proposed protocol has the adaptively UC-security, stored in a file server which locates behind the company which makes the adversary forge signatures of previous gateway. To access the file, someone outside the periods more difficultly even though she can corrupt company should launch the successful authentication users at any time. The proposed protocol shares the with a correct password between himself and the secret with the additional sharing. The size of the authentication gateway. However the file is so sensitive signing key is constant and the size of the partial Corresponding Author: Xuan Hong, Department of Computer Science and Technology, Shanghai Normal University, 100 Guilin Road, Shanghai 200234, China 270 Res. J. Appl. Sci. Eng. Technol., 5(1): 270-274, 2013 signature is constant, which are independent to the the signer is corrupted and FSATSig is asked to verify (m, number of the users. Furthermore, the threshold s), then FSATSig allows the ideal-process adversary to signature generation stage and the signature combining force the answer to be “1”, even if m was never before stage are completely non-interactive. At last, the signed. We add the corrupted signer and make the correctness and security have been analyzed. whole framework more reasonable. The functionality FSATSig is also the standard UNIVERSAL COMPOSABLE SECURE MODEL corruption functionality, with an additional stipulation. OF SERVER-AIDED THRESHOLD When a user P is corrupted, FSATSig records this fact and reports to the adversary all the signing and verification Recently some efforts were made towards requests made by P. If P is the signer then the adversary capturing the security requirements within the universal gets also the signature share generation algorithm SS composability framework. This modeling has some together with its current state. The verification significant advantages in designing and analyzing algorithm V is deterministic guarantees consistency, complex system. The framework maintains the security, namely that all verification requests for the same triple regardless of execution environment. That is suitable return the same answer. Verifying that V (m, σ) = 1 at for analyzing the mobile ad-hoc networks. Canetti signature generation time guarantees completeness, introduced the universal composability framework namely that if a signature was generated legally then it (Canetti and Rabin, 2003; Canetti, 2004) as the new will pass verification. The check for forgery at approach for designing and analyzing the security of verification time guarantees unforgeability. cryptographic primitives and protocols. The main concern is to create new approach for the assessment of Proxy threshold signature F : cryptographic protocol. It is guaranteed that a protocol SATSig with UC security maintains its security even when Setup: When received (Setup, sid) from party P, running concurrently with others. When analyzing multiprotocol systems, we analyze each protocol as if it where sid = (P, sid’), FSATSig sends (Setup, sid) to is stand alone Then the composition theorem is used to the adversary A. When received (Algorithms, sid, deduce the security of all instances when running SS, SV, SC, V) from the adversary A, where SS is concurrently. description of a PPT ITM and SV, SC, V are Security of the protocol π is achieved via description of the deterministic PPT ITMs. FSATSig comparing the protocol execution in the real-life model sends (VeriAlgo, sid, SV) to the party P and to the ideal model. This is formalized by considering an (VeriAlgo, sid, V) to the server. environment Z, which represents all the other protocols Signature share: When received (ThSign, sid, m) running in the system, including the protocols that from the party P, where sid = (P, sid’), FSATSig sets provide inputs to and obtain outputs from. Security is σ = SS (m), records the entry (m, σ), sends required that comparing protocol π with an ideal (ThSignature, sid, m, σ) to party P. functionality F, that is, whatever Z could achieve by Combine: When received (Combine, sid, m, σ1, σ2, attacking π, it could also achieve by interacting with F. …, σt) from the server S, FSATSig check whether all in other words, if no environment can distinguish the t shares are registered. If so, FSATSig lets σ = SC interactions with π from those with F, the protocol π is (σ1,…,σt), records the entry (m, σ), outputs secure in the hybrid model. To make this precise, a (Combined, sid, m, σ) to S. special ITM S is introduced. The goal of S is to Verify: When received (Verify, sid, m, σ) from simulate the adversary’s view of π, based on the party P, output (Verified, sid, m, σ, f)) to P, where information is willing to exchange with environment. f is determined as follows: We define the functionality FSATSig for the server- aided threshold signature, whose basic ideal is o If V’ = V and the entry (m, σ) is recorded, then set f providing a registry service" where the users can = 1. register their (message, share) pairs. Any party that o Else, if V’ = V and no entry (m, σ) is recorded, then provides the right verification key can check whether a set f = 0. given pair is registered. It takes four types of inputs, o Else, if V’ ≠ V, then set f = 0. which correspond to the four basic modules of the signature protocol: Setup, signature share, combine and SERVER-AIDED THRESHOLD SIGNATURE verify. PROTOCOL We also deal with the case that the signer is corrupted. Corrupted signers are controlled by the We present the server assisted threshold RSA adversary, which is different from the honest ones. If signature protocol π, which realizes FSATSig in a 271 Res. J. Appl. Sci. Eng. Technol., 5(1): 270-274, 2013 straightforward way. All parties in the protocol run zskcr i the following code: Combine: When the server receives the t signature Setup: It is run by the Key Generation Center (KGC). shares from some t signers. It firstly checks whether Given the security parameter k, the KGC perform RSA these shares are valid: key generation with secure parameter k to obtain modulus N, where, ? 4242z zc cHvxv'( , ,i , , vx , ) Npqpp, 2'1, q 2'1 q If the verification equations hold simultaneously, the with p, q, p’, q’ themselves prime.
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