Fault-Tolerant Parallel Integer Multiplication

Exascale machines have a small mean time between failures, necessitating fault tolerance. Out-of-the-box fault-tolerant solutions, such as checkpoint-restart and replication, apply to any algorithm but incur significant overhead costs. Long integer multiplication is a fundamental kernel in numerical linear algebra and cryptography. The na ve, schoolbook multiplication algorithm runs inΘ ( n2k while Toom-Cook algorithms runs in Θ ( nlogκ (2κ-1) for 2 ≤ κ. We obtain the first efficient fault-tolerant parallel Toom-Cook algorithm. While asymptotically faster FFT-based algorithms exist, Toom-Cook algorithms are often favored in practice on small scale and on supercomputers. Our algorithm enables fault tolerance with negligible overhead costs. Compared to existing, general-purpose, faulttolerant solutions, our algorithm reduces the arithmetic and communication (bandwidth) overhead costs by a factor of Θ P (2κ-1) (where P is the number of processors). To this end, we adapt the fault-tolerant BFS-DFS method of Birnbaum et al. (2020) for fast matrix multiplication and combine it with a coding strategy tailored for Toom-Cook. This eliminates the need for recomputations, resulting in a much faster algorithm..