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Rainbow tables are a set of pre-computed lookup tables used to significantly accelerate a brute-force attack of several specific cryptosystems. In cryptography, a brute-force attack is a process of deriving a cryptographic key by trying every possible combination within a specific keyspace until the correct one is found. How quickly this key can be found depends on the size of the key space, and the computing resources applied. 

 

A 40-bit cryptosystem has slightly more than one trillion keys (2⁴⁰ = 1.099 trillion). If a single computer can test 500,000 keys per second, this computer would need approximately 25 days to exhaust the key space. A rainbow table improves the efficiency of the brute force attack by having all possibilities precomputed and saved from the start. As a result, a single computer can break a 40-bit encrypted file in slightly more than a minute rather than in weeks.