Message Authentication Codes (MACs) used in today's wireless communication standards may not be able to satisfy resource limitations of simpler 5G radio types and use cases such as machine type communications. As a possible solution, we present a lightweight message authentication scheme based on the cyclic redundancy check (CRC). It has been previously shown that a CRC with an irreducible generator polynomial as the key is an -almost XOR-universal (AXU) hash function with = (m + n)/2n-1, where m is the message size and n is the CRC size. While the computation of n-bit CRCs can be efficiently implemented in hardware using linear feedback shift registers, generating random degree-n irreducible polynomials is computationally expensive for large n. We propose using a product of k irreducible polynomials whose degrees sum up to n as a generator polynomial for an n-bit CRC and show that the resulting hash functions are -AXU with = (m + n)k/2n -k. The presented message authentication scheme can be seen as providing a trade-off between security and implementation efficiency.