If 
is an arbitrary integer relatively prime to 
and 
is a primitive root of 
, then there exists among the numbers
0, 1, 2, ..., 
,
where 
is the totient function, exactly
one number 
such that
 |
The number 
is then called the discrete logarithm of 
with respect to the base 
modulo 
and is denoted
 |
The term "discrete logarithm" is most commonly used in cryptography, although the term "generalized multiplicative order" is sometimes used as well (Schneier 1996, p. 501). In number theory, the term "index" is generally used instead (Gauss 1801; Nagell 1951, p. 112).
For example, the number 7 is a positive primitive root of 
(in fact, the set of primitive roots of 41 is given by 6, 7, 11, 12, 13, 15, 17,
19, 22, 24, 26, 28, 29, 30, 34, 35), and since 
, the number 15 has multiplicative order 3 with
respect to base 7 (modulo 41) (Nagell 1951, p. 112). The generalized multiplicative
order is implemented in the Wolfram Language
as MultiplicativeOrder[g,
n, 
a1
], or more generally as MultiplicativeOrder[g,
n, 
a1,
a2, ...
].
Discrete logarithms were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight" of the television crime drama NUMB3RS.
" and "The Index Calculus." §31 and 33
in