subexponentialtime

Subexponential time refers to a broad family of algorithmic running times that grow faster than any polynomial in the input size but slower than true exponential time. If n denotes the size of the input (typically measured in bits), an algorithm runs in subexponential time when its worst-case running time is exp(o(n)) or equivalently 2^{o(n)}. This captures bounds that are between polynomial and exponential growth.

Subexponential bounds are often described using the L-notation: L_n[α, c] = exp((c+o(1))(log n)^α (log log n)^{1−α}) for α in [0,1] and c>0. For many number-theoretic problems, α=1/3 gives a typical subexponential form. Thus, while not polynomial, these algorithms are significantly faster than brute-force exponential methods for large inputs.

Prominent examples include factoring and discrete logarithm problems. The General Number Field Sieve (GNFS) for factoring an integer N runs in time roughly exp((c+o(1))(log N)^{1/3}(log log N)^{2/3}), which is subexponential in the number of bits of N. Index calculus methods for discrete logarithms in prime fields exhibit similar subexponential behavior. By contrast, problems such as the elliptic curve discrete logarithm have algorithms whose best known running times are exponential in the input size, illustrating that subexponential performance is problem-dependent.

In practice, subexponential time bounds are important in computational number theory and cryptography, where they inform security assessments and the feasibility of algorithms for large instances. They describe a middle ground: faster-than-exponential but not polynomial, shaping both theoretical analyses and practical expectations.