2-Dimensional interval algorithm

The interval algorithm by Han and Hoshi is an efficient algorithm which can convert a sequence of random variables into another sequence of random variable with a required probability distribution. In this paper, we extend the interval algorithm to transform a pair of sequences of random variable into another pair of sequences of random variables. The extension allows some independency or functional dependency constraints between the input and output sequences. The possible applications in key extraction and random number generation are discussed. The proposed algorithm can asymptotically achieve the optimal output rate and it minimizes the length of the input sequence required to generate the output sequence for certain input distributions.