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We present a new method for proving strong lower bounds in communication complexity. This method is based on the notion of the conditional information complexity of a function which is the minimum amount of information about the inputs that has to be revealed by a communication protocol for the function. While conditional information complexity is a lower bound on the communication complexity, we show that it also admits a direct sum theorem. Direct sum decomposition reduces our task to that of proving (conditional) information complexity lower bounds for simple problems (such as the AND of two bits). For the latter, we develop novel techniques based on Hellinger distance and its generalizations.