Editing Euler's totient function (section)

===The RSA cryptosystem===

{{main article|RSA (algorithm)}}

Setting up an RSA system involves choosing large prime numbers {{mvar|p}} and {{mvar|q}}, computing {{math|''n'' {{=}} ''pq''}} and {{math|''k'' {{=}} ''φ''(''n'')}}, and finding two numbers {{mvar|e}} and {{mvar|d}} such that {{math|''ed'' ≡ 1 (mod ''k'')}}. The numbers {{mvar|n}} and {{mvar|e}} (the "encryption key") are released to the public, and {{mvar|d}} (the "decryption key") is kept private.

A message, represented by an integer {{mvar|m}}, where {{math|0 < ''m'' < ''n''}}, is encrypted by computing {{math|''S'' {{=}} ''m''<sup>''e''</sup> (mod ''n'')}}.

It is decrypted by computing {{math|''t'' {{=}} ''S''<sup>''d''</sup> (mod ''n'')}}. Euler's Theorem can be used to show that if {{math|0 < ''t'' < ''n''}}, then {{math|''t'' {{=}} ''m''}}.

The security of an RSA system would be compromised if the number {{mvar|n}} could be efficiently factored or if {{math|''φ''(''n'')}} could be efficiently computed without factoring {{mvar|n}}.