Özet:
The term \Extensions" in the thesis title refers to homomorphism and veri ability additions to the Asmuth-Bloom scheme. Homomorphic abilities enable computations on hidden data without opening it, and veri ability eliminates the necessity of a trusted third party. Both abilities jointly facilitate secure multi-party computation. Multi-party computation has became one of the main research areas of the crypto-community with the goal to create a protocol to jointly compute a function using their inputs without revealing anything but the result. With the technological developments, the demand for personal data storage and computation have increased over the last decades. In order to maintain computational operations over the data, it is usually stored without encryption which brings along some privacy concerns. Rivest et al. proposed homomorphic encryption to overcome privacy issues, while keeping the functionality. After more than a quarter century, in 2009, Craig Gentry proposed the rst fully homomorphic encryption scheme, and security of his scheme relies on an assumption of the hardness of a mathematical problem, i.e. the approximate GCD problem. Nonetheless, unconditionally or information-theoretically secure computation can be done by secret sharing schemes. In this thesis, we explore homomorphic properties of a well-known secret sharing scheme: Asmuth-Bloom scheme. We propose several modi ed versions having homomorphic properties with their security analysis. Another important contribution of the thesis is related to Asmuth-Bloom based veri able secret sharing. First, we analyse the existing schemes and expose their weaknesses. Secondly, we propose the rst veri able secret sharing scheme secure against unbounded adversaries, and we apply this scheme to construct joint random secret sharing scheme.
