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[ci skip] It is *not* the case that a discrete log exists when the base and modulus are coprime. Take 4^x = 2 mod 5 as a counterexample. Change API accordingly.

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Nick Thompson
2018-02-10 17:51:59 -06:00
parent 4f4f3eda37
commit faa61cd911
3 changed files with 50 additions and 48 deletions
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10
doc/modular_arithmetic/discrete_log.qbk
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@ -29,7 +29,7 @@ The algorithms provided by Boost should be acceptable up to roughly 32 bits.
public:
bsgs_discrete_log(Z base, Z p);
Z operator()(Z arg) const;
boost::optional<Z> operator()(Z arg) const;
};
}}
@ -49,7 +49,7 @@ Basic usage is shown below:
}
auto log_2 = bsgs_discrete_log(2, 5);
int log = log_2(3);
int log = log_2(3).value();
std::cout << "log_2(3) mod 5 = " << log << std::endl;
@ -78,10 +78,10 @@ The baby-step giant-step algorithm is less polite when the base and the modulus
}
The baby-step giant-step discrete log will *never* compute a logarithm when the generator and modulus are not coprime,
The baby-step giant-step discrete log will *never* compute a logarithm when the base and modulus are not coprime,
because it relies on the existence of modular multiplicative inverses.
However, discrete logarithms can exist even when the generator and modulus share a common divisor greater than 1.
For example, since 2[sup 1] = 2 mod 4, log[sub 2](2) = 1.
However, discrete logarithms can exist even when the base and modulus share a common divisor greater than 1.
For example, since 2[super 1] = 2 mod 4, log[sub 2](2) = 1.
Trial multiplication successfully recovers this value, and `bsgs_discrete_log` blows up.