Editing Modular exponentiation (section)

== Direct method ==
The most direct method of calculating a modular exponent is to calculate {{math|''b''<sup>''e''</sup>}} directly, then to take this number modulo {{math|''m''}}.  Consider trying to compute {{math|''c''}}, given {{math|''b'' {{=}} 4}}, {{math|''e'' {{=}} 13}}, and {{math|''m'' {{=}} 497}}:
: {{math|''c'' ≡ 4{{sup|13}} (mod 497)}}

One could use a calculator to compute 4<sup>13</sup>; this comes out to 67,108,864.  Taking this value modulo 497, the answer {{math|''c''}} is determined to be 445.

Note that {{math|''b''}} is only one digit in length and that {{math|''e''}} is only two digits in length, but the value {{math|''b''<sup>''e''</sup>}} is 8 digits in length.

In strong cryptography, {{math|''b''}} is often at least 1024 [[bit]]s.<ref>{{Cite web|url=https://weakdh.org/|title=Weak Diffie–Hellman and the Logjam Attack|website=weakdh.org|access-date=2019-05-03}}</ref>  Consider {{math|''b'' {{=}} 5 × 10<sup>76</sup>}} and {{math|''e'' {{=}} 17}}, both of which are perfectly reasonable values.  In this example, {{math|''b''}} is 77 digits in length and {{math|''e''}} is 2 digits in length, but the value {{math|''b''<sup>''e''</sup>}} is 1,304 decimal digits in length.  Such calculations are possible on modern computers, but the sheer magnitude of such numbers causes the speed of calculations to drop considerably.  As {{math|''b''}} and {{math|''e''}} increase even further to provide better security, the value {{math|''b''<sup>''e''</sup>}} becomes unwieldy.

The time required to perform the exponentiation depends on the operating environment and the processor.  The method described above requires {{math|[[Big O notation|Θ]](''e'')}} multiplications to complete.