Generalised Cumulative Arrays in Secret Sharing

Authors:
Shoulun Long;Josef Pieprzyk;Huaxiong Wang;Duncan S. Wong
Affiliations:
Department of Mathematics, University of Science and Technology of China, Hefei, China;Department of Computing, Centre for Advanced Computing --- Algorithms and Cryptography, Macquarie University, Sydney, Australia 2109;Department of Computing, Centre for Advanced Computing --- Algorithms and Cryptography, Macquarie University, Sydney, Australia 2109;Department of Computer Science, City University of Hong Kong, Hong Kong, China
Venue:
Designs, Codes and Cryptography
Year:
2006

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Abstract

Cumulative arrays have played an important role in the early development of the secret sharing theory. They have not been subject to extensive study so far, as the secret sharing schemes built on them generally result in much larger sizes of shares, when compared with other conventional approaches. Recent works in threshold cryptography show that cumulative arrays may be the appropriate building blocks in non-homomorphic threshold cryptosystems where the conventional secret sharing methods are generally of no use. In this paper we study several extensions of cumulative arrays and show that some of these extensions significantly improve the performance of conventional cumulative arrays. In particular, we derive bounds on generalised cumulative arrays and show that the constructions based on perfect hash families are asymptotically optimal. We also introduce the concept of ramp perfect hash families as a generalisation of perfect hash families for the study of ramp secret sharing schemes and ramp cumulative arrays.