Abstract: A secure multi-party computing system performs a multi-pivot partial sorting operation on a secret shared array of values. The use of multiple pivots supports efficient computations in a multi-party computation setting. Partial sorting determines percentile values without the need for a full sort. The secret shared array is first permuted by a secret random permutation. A multi-pivot sort, which can be a partial sort, is performed on the permuted array to obtain a public sorting permutation. The multi-pivot sort uses oblivious comparisons that produce secret shared Boolean indications of whether one secret shared value is less than another. The Boolean indications are revealed and used to produce the public sorting permutation, which in turn, is applied to the secret random permutation to obtain a secret shared sorting permutation. The secret shared sorting permutation is then applied to the secret shared array to obtain a sorted secret shared result.

Abstract: An oblivious comparison method takes as input two secret shared numerical values x and y and outputs a secret shared bit that is the result of the comparison of x and y (e.g. 1 if x<y and 0 otherwise). The method uses secure multi-party computation, allowing multiple parties to collaboratively perform the comparison while keeping the inputs private and revealing only the result. The two secret shared values are subtracted to compute a secret shared result, the sign of which indicates the result of the comparison. The method decomposes the secret shared result into a masked Boolean representation and then performs a bit-wise addition of the mask and the masked result. Through the bit-wise addition the method can extract a secret shared representation of the most significant bit, which indicates the sign of the result, without revealing the result itself.