In modern cryptography, the problem of secure multiparty computation is about the cooperation between mutually distrusting parties computing a given function. Each party holds some private information that should remain secret as much as possible throughout the computation. A large body of research initiated in the early 1980's has shown that any computable function can be evaluated using secure multiparty computation. Though these feasibility results are general, their applicability in practical situations is rather unsatisfactory. This thesis concerns the study of two particular cryptographic primitives with focus on efficiency. The first primitive studied is a generalization of verifiable shuffles of homomorphic encryptions, where the shuffler is only allowed to apply a permutation from a restricted set of permutations. In this thesis, we consider shuffles using permutations from a k-fragile set, meaning that any k input-output correspondences uniquely identify a permutation within the set. We provide verifiable shuffles restricted to the set of all rotations (1-fragile), affine transformations (2-fragile), and Möbius transformations (3-fragile). Applications of these special shuffles include fragile mixing, electronic elections, secure function evaluation using scrambled circuits, and secure integer comparison. Two approaches for verifiable rotations are presented. On the one hand, we use properties of the Discrete Fourier Transform (DFT) to express in a compact way that a rotation is applied in a shuffle. The solution is efficient, but imposes some mild restrictions on the parameters to allow DFT to work. On the other hand, we present a general solution that does not impose any parameter constraint and works on any homomorphic cryptosystem. These protocols for rotations are used to build efficient shuffling protocols for affine and Möbius transformations. The second primitive is secure integer comparison. In a general scenario, parties are given homomorphic encryptions of the bits of two integers and, after running a protocol, an encryption of a bit is produced, telling the result of the greater-than comparison of the two integers. This is a useful building block for higher-level protocols such as electronic voting, biometrics authentication or electronic auctions. A study of the relationship of other problems to integer comparison is given as well. We present two types of solutions for integer comparison. Firstly, we consider an arithmetic circuit yielding secure protocols within the framework for multiparty computation based on threshold homomorphic cryptosystems. Our circuit achieves a good balance between round and computational complexities, when compared to the similar solutions in the literature. The second type of solutions uses a intricate approach where different building blocks are used. A full analysis is made for the two-party case where efficiency of the resulting protocols compares favorably to other solutions and approaches
