This paper describes a novel method for efficiently calculating CRC checksums without lookup tables or hardware support for polynomial multiplication. Throughput of CRC32 is increased by 100% across different platforms compared with the current state of the art. Performance is on par with or exceeds hardware-accelerated solutions on x86_64 and ARMv8 processors, and these hardware-accelerated solutions see a performance increase of 5-20% depending on message length. The small number of operations required with this approach could simplify hardware CRC32 implementations.