We consider the setting of a Master server, M , who possesses confidential data and wants to run intensive computations on it, as part of a machine learning algorithm for example. The Master wants to distribute these computations to untrusted workers who volunteered to help with this task. However, the data must be kept private in an information theoretic sense. Some of the workers may be stragglers, e.g., slow or busy. We are interested in reducing the delays experienced by the Master. We focus on linear computations as an essential operation in many iterative algorithms. We propose a solution based on new codes, called Staircase codes, introduced previously by two of the authors. Staircase codes allow flexibility in the number of stragglers up to a given maximum, and universally achieve the information theoretic limit on the download cost by the Master, leading to latency reduction. We find upper and lower bounds on the Master's mean waiting time. We derive the distribution of the Master's waiting time, and its mean, for systems with up to two stragglers. We show that Staircase codes always outperform existing solutions based on classical secret sharing codes. We validate our results with extensive implementation on Amazon EC2.