Rainbow tables partition a function, whose domain is a set of values and whose codomain is a set of keys derived from those values, into chains such that each chain is an alternating sequence of values and keys, followed by a final value.
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Rainbow tables were invented by Philippe Oechslin as an application of an earlier, simpler algorithm by Martin Hellman.
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Rainbow tables are a special kind of such table that overcome certain technical difficulties.
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Rainbow tables are much bigger and use a different reduction function in each column.
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Rainbow tables effectively solve the problem of collisions with ordinary hash chains by replacing the single reduction function R with a sequence of related reduction functions R1 through Rk.
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Rainbow tables can achieve similar performance with tables that are k times larger, allowing them to perform a factor of k fewer lookups.
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Rainbow tables use a refined algorithm with a different reduction function for each "link" in a chain, so that when there is a key collision in two or more chains, the chains will not merge as long as the collision doesn't occur at the same position in each chain.
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Rainbow tables table is ineffective against one-way hashes that include large salts.
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However, Rainbow tables can be generated that take into account common ways in which users attempt to choose more secure passwords, such as adding a number or special character.
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Rainbow tables have seen reduced usage as of 2020 as salting is more common and GPU-based brute force attacks have become more practical.
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However, rainbow tables are available for eight and nine character NTLM passwords.
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