On dealer-free dynamic threshold schemes

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February 2013, 7(1): 39-56. doi: 10.3934/amc.2013.7.39

On dealer-free dynamic threshold schemes

Mehrdad Nojoumian ^1, and Douglas R. Stinson ^2,

1.	Department of Computer Science, Southern Illinois University, Carbondale, IL 62901, United States
2.	David R. Cheriton School of Computer Science, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Received April 2012 Published January 2013

Abstract
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In a threshold scheme, the sensitivity of the secret as well as the number of players may fluctuate due to various reasons, e.g., mutual trust may vary or the structure of the players' organization might be changed. A possible solution to this problem is to modify the threshold and/or change the secret. Moreover, a common problem with almost all secret sharing schemes is that they are "one-time", meaning that the secret and shares are known to everyone after a public secret recovery process. This problem could be resolved if the dealer shares various secrets at the beginning, but a better solution is to dynamically generate new secrets in the absence of the dealer. These issues are our main motivation to revisit dynamic threshold schemes.
Therefore, we first provide the first comprehensive study of threshold modification techniques in both the passive and active adversary models. We first review an existing method for threshold modification based on resharing shares of a secret; this method is secure in the setting of a passive adversarial coalition. We then discuss two methods, termed public evaluation (for threshold reduction) and zero addition (for threshold increase) that can be used in both the passive and active adversarial setting. In the case of an active adversary, the techniques make use of verifiable secret sharing schemes, whereas the schemes considered in the passive adversary model are all based on the Shamir scheme. As an application, we discuss how the threshold and the secret can be changed multiple times to arbitrary values after the scheme's initialization.

Keywords: unconditional security, threshold and secret changeability., Secret sharing.

Mathematics Subject Classification: Primary: 94A6.

Citation: Mehrdad Nojoumian, Douglas R. Stinson. On dealer-free dynamic threshold schemes. Advances in Mathematics of Communications, 2013, 7 (1) : 39-56. doi: 10.3934/amc.2013.7.39

References:

[1]	S. G. Barwick, W. A. Jackson and K. M. Martin, Updating the parameters of a threshold scheme by minimal broadcast,, IEEE Trans. Inform. Theory, 51 (2005), 620. doi: 10.1109/TIT.2004.840857.
[2]	D. Beaver, Multiparty protocols tolerating half faulty processors,, in, (1989), 560.
[3]	M. Ben-Or, S. Goldwasser and A. Wigderson, Completeness theorems for non-cryptographic fault-tolerant distributed computation,, in, (1988), 1.
[4]	B. Blakley, G. R. Blakley, A. H. Chan and J. L. Massey, Threshold schemes with disenrollment,, in, (1992), 540.
[5]	G. R. Blakley, Safeguarding cryptographic keys,, in, (1979), 313.
[6]	C. Blundo, A. Cresti, A. De Santis and U. Vaccaro, Fully dynamic secret sharing schemes,, Theoret. Comp. Sci., 165 (1996), 407.
[7]	B. Chor, S. Goldwasser, S. Micali and B. Awerbuch, Verifiable secret sharing and achieving simultaneity in the presence of faults,, in, (1985), 383.
[8]	P. D'Arco and D. R. Stinson, On unconditionally secure robust distributed key distribution centers,, in, (2002), 346.
[9]	Y. Desmedt and S. Jajodia, Redistributing secret shares to new access structures and its applications,, in, (1997), 97.
[10]	R. Gennaro, Y. Ishai, E. Kushilevitz and T. Rabin, The round complexity of verifiable secret sharing and secure multicast,, in, (2001), 580.
[11]	R. Gennaro, M. O. Rabin and T. Rabin, Simplified vss and fast-track multiparty computations with applications to threshold cryptography,, in, (1998), 101.
[12]	A. Herzberg, S. Jarecki, H. Krawczyk and M. Yung, Proactive secret sharing or: How to cope with perpetual leakage,, in, (1995), 339.
[13]	I. Ingemarsson and G. J. Simmons, A protocol to set up shared secret schemes without the assistance of a mutualy trusted party,, in, (1990), 266.
[14]	W.-A. Jackson, K. M. Martin and C. M. O'Keefe, Mutually trusted authority-free secret sharing schemes,, J. Cryptology, 10 (1997), 261. doi: 10.1007/s001459900031.
[15]	A. Maeda, A. Miyaji and M. Tada, Efficient and unconditionally secure verifiable threshold changeable scheme,, in, (2001), 403.
[16]	K. Martin, Dynamic access policies for unconditionally secure secret sharing schemes,, in, (2005), 61.
[17]	K. M. Martin, J. Pieprzyk, R. Safavi-Naini and H. X. Wang, Changing thresholds in the absence of secure channels,, in, (1999), 177.
[18]	K. M. Martin, R. Safavi-Naini and H. X. Wang, Bounds and techniques for efficient redistribution of secret shares to new access structures,, Computer J., 42 (1999), 638.
[19]	V. Nikov and S. Nikova, On proactive secret sharing schemes,, in, (2004), 308.
[20]	M. Nojoumian, D. R. Stinson and M. Grainger, Unconditionally secure social secret sharing scheme,, IET Inform. Secur., 4 (2010), 202. doi: 10.1049/iet-ifs.2009.0098.
[21]	T. Rabin and M. Ben-Or, Verifiable secret sharing and multiparty protocols with honest majority,, in, (1989), 73.
[22]	A. Shamir, How to share a secret,, Commun. ACM, 22 (1979), 612. doi: 10.1145/359168.359176.
[23]	R. Steinfeld, H. X. Wang and J. Pieprzyk, Lattice-based threshold-changeability for standard shamir secret-sharing schemes,, in, (2004), 170.
[24]	D. R. Stinson and R. Z. Wei, Unconditionally secure proactive secret sharing scheme with combinatorial structures,, in, (1999), 200.
[25]	C. Tartary and H. X. Wang, Dynamic threshold and cheater resistance for shamir secret sharing scheme,, in, (2006), 103. doi: 10.1007/11937807_9.

show all references

References:

[1]	S. G. Barwick, W. A. Jackson and K. M. Martin, Updating the parameters of a threshold scheme by minimal broadcast,, IEEE Trans. Inform. Theory, 51 (2005), 620. doi: 10.1109/TIT.2004.840857.
[2]	D. Beaver, Multiparty protocols tolerating half faulty processors,, in, (1989), 560.
[3]	M. Ben-Or, S. Goldwasser and A. Wigderson, Completeness theorems for non-cryptographic fault-tolerant distributed computation,, in, (1988), 1.
[4]	B. Blakley, G. R. Blakley, A. H. Chan and J. L. Massey, Threshold schemes with disenrollment,, in, (1992), 540.
[5]	G. R. Blakley, Safeguarding cryptographic keys,, in, (1979), 313.
[6]	C. Blundo, A. Cresti, A. De Santis and U. Vaccaro, Fully dynamic secret sharing schemes,, Theoret. Comp. Sci., 165 (1996), 407.
[7]	B. Chor, S. Goldwasser, S. Micali and B. Awerbuch, Verifiable secret sharing and achieving simultaneity in the presence of faults,, in, (1985), 383.
[8]	P. D'Arco and D. R. Stinson, On unconditionally secure robust distributed key distribution centers,, in, (2002), 346.
[9]	Y. Desmedt and S. Jajodia, Redistributing secret shares to new access structures and its applications,, in, (1997), 97.
[10]	R. Gennaro, Y. Ishai, E. Kushilevitz and T. Rabin, The round complexity of verifiable secret sharing and secure multicast,, in, (2001), 580.
[11]	R. Gennaro, M. O. Rabin and T. Rabin, Simplified vss and fast-track multiparty computations with applications to threshold cryptography,, in, (1998), 101.
[12]	A. Herzberg, S. Jarecki, H. Krawczyk and M. Yung, Proactive secret sharing or: How to cope with perpetual leakage,, in, (1995), 339.
[13]	I. Ingemarsson and G. J. Simmons, A protocol to set up shared secret schemes without the assistance of a mutualy trusted party,, in, (1990), 266.
[14]	W.-A. Jackson, K. M. Martin and C. M. O'Keefe, Mutually trusted authority-free secret sharing schemes,, J. Cryptology, 10 (1997), 261. doi: 10.1007/s001459900031.
[15]	A. Maeda, A. Miyaji and M. Tada, Efficient and unconditionally secure verifiable threshold changeable scheme,, in, (2001), 403.
[16]	K. Martin, Dynamic access policies for unconditionally secure secret sharing schemes,, in, (2005), 61.
[17]	K. M. Martin, J. Pieprzyk, R. Safavi-Naini and H. X. Wang, Changing thresholds in the absence of secure channels,, in, (1999), 177.
[18]	K. M. Martin, R. Safavi-Naini and H. X. Wang, Bounds and techniques for efficient redistribution of secret shares to new access structures,, Computer J., 42 (1999), 638.
[19]	V. Nikov and S. Nikova, On proactive secret sharing schemes,, in, (2004), 308.
[20]	M. Nojoumian, D. R. Stinson and M. Grainger, Unconditionally secure social secret sharing scheme,, IET Inform. Secur., 4 (2010), 202. doi: 10.1049/iet-ifs.2009.0098.
[21]	T. Rabin and M. Ben-Or, Verifiable secret sharing and multiparty protocols with honest majority,, in, (1989), 73.
[22]	A. Shamir, How to share a secret,, Commun. ACM, 22 (1979), 612. doi: 10.1145/359168.359176.
[23]	R. Steinfeld, H. X. Wang and J. Pieprzyk, Lattice-based threshold-changeability for standard shamir secret-sharing schemes,, in, (2004), 170.
[24]	D. R. Stinson and R. Z. Wei, Unconditionally secure proactive secret sharing scheme with combinatorial structures,, in, (1999), 200.
[25]	C. Tartary and H. X. Wang, Dynamic threshold and cheater resistance for shamir secret sharing scheme,, in, (2006), 103. doi: 10.1007/11937807_9.

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