Provided is a method and system for producing message authentication tags and a method and system for producing hash values using bit-mixers. The methods include producing a message authentication or hash value by obtaining a message; segmenting, padding by an electronic processor, the message into a number of equal sized message blocks comprising a first message block, one or more subsequent message blocks, and a final message block; performing, by the electronic processor, a first bit-mixing operation on the first message block with an initialization value; performing, by the electronic processor, subsequent bit-mixing operations on the one or more subsequent message blocks and the final message block with a previous message block; and producing, by the electronic processor, the message authentication tag based on the first bit-mixing operation and the subsequent bit-mixing operations, employing a secret key material.