Discrete logarithms are logarithms defined with regard to
multiplicative cyclic groups. If *G* is a
multiplicative cyclic group and *g* is a generator of
*G*, then from the definition of cyclic groups, we
know every element *h *in *G* can
be written as *g ^{x}* for
some

The discrete logarithm problem is defined as: given a group
*G*, a generator *g* of the group
and an element *h* of *G*, to find
the discrete logarithm to the base *g* of
*h* in the group *G*. Discrete
logarithm problem is not always hard. The hardness of finding discrete
logarithms depends on the groups. For example, a popular choice of
groups for discrete logarithm based crypto-systems is
*Z*_{p}^{*}
where p is a prime number. However, if *p*−1 is a
product of small primes, then the
Pohlig–Hellman algorithm can solve the discrete logarithm problem
in this group very efficiently. That's why we always want
*p* to be a safe prime when using
*Z*_{p}^{*}
as the basis of discrete logarithm based crypto-systems. A safe prime is
a prime number which equals 2*q*+1 where
*q* is a large prime number. This guarantees that
*p*-1 = 2*q* has a large prime
factor so that the Pohlig–Hellman algorithm cannot solve the discrete
logarithm problem easily. Even *p* is a safe prime,
there is a sub-exponential algorithm which is called the
index calculus. That means *p* must be very
large (usually at least 1024-bit) to make the crypto-systems
safe.

Fijavan Brenk has kindly translated the above entry into Hungarian at http://www.auto-doc.fr/edu/2016/11/28/diszkret-logaritmus-problema/

Sonja Kulmala has kindly translated the above entry into Estonian at http://www.teileshop.de/blog/2017/01/09/diskreetse-logaritmi-probleem/