DLOG

DLOG is commonly used to refer to the discrete logarithm problem in number theory and cryptography. In a finite cyclic group G with a chosen generator g, the problem is: given an element h = g^x, find the exponent x. The value x is called the discrete logarithm of h to the base g, often written log_g(h).

A typical setting is the multiplicative group of integers modulo a prime p, denoted GF(p)^× or Z_p^×. There, x satisfies g^x ≡ h (mod p) for known g and h. The same idea extends to other groups, including elliptic curve groups, where the operation is defined by the group law on the curve. The discrete logarithm in a given group may be easy or hard depending on the group’s structure and parameter size.

Security and cryptographic relevance are central to DLOG. The difficulty of solving the discrete logarithm problem underpins several widely used protocols, including the Diffie–Hellman key exchange and the Digital Signature Algorithm. Security levels depend on the chosen group and parameters; certain groups admit subexponential-time algorithms that reduce effective security if parameters are not large enough.

Algorithms and complexity vary by group. For general finite fields, methods such as index calculus can solve DLOG in subexponential time. In contrast, for elliptic curve groups, the best known algorithms are generic and rely on roughly the square root of the group order, allowing comparable security with much smaller key sizes.

Note that DLOG as an acronym can appear in other domains, but in mathematical and cryptographic contexts it most often denotes the discrete logarithm problem. See also discrete logarithm, Diffie–Hellman, and elliptic-curve cryptography.