ÐIn their 1982 paper, Adams and Siegel proposed an Extra Stage Cube Interconnection Network that tolerates one switch failure with one extra stage. We extend their results and discover a class of Extra Stage Interconnection Networks that tolerate multiple switch failures with a minimal number of extra stages. Adopting the same fault model as Adams and Siegel, the faulty switches can be bypassed by a pair of demultiplexer/multiplexer combinations. It is easy to show that, to maintain point to point and broadcast connectivities, there must be at least f extra stages to tolerate f switch failures. We present the first known construction of an Extra Stage Interconnection Network that meets this lower-bound. This n-dimensional Multistage Interconnection Network has n f stages and tolerates f switch failures. An n-bit label called mask is used for each stage that indicates the bit differences between the two inputs coming into a common switch. We designed the fault-tolerant construction such that it repeatedly uses the singleton basis of the n-dimensional vector space as the stage mask vectors. This construction is further generalized and we prove that an n-dimensional Multistage Interconnection Network is optimally fault-tolerant if and only if the mask vectors of every n consecutive stages span the n-dimensional vector space. Index TermsÐMultistage Interconnection Networks (MIN), fault tolerance, extrastage, switch faults, stage masks.