largeinteger

A large integer is an integer whose value cannot be represented by a fixed-size machine word alone. In modern software, numbers larger than 32 bits or 64 bits require multi-precision or arbitrary-precision arithmetic to be stored and manipulated exactly. Large integers are essential in areas such as cryptography, number theory, and symbolic computation, where exact results and very large values are common.

Representation typically involves sign information and a sequence of digits or limbs. The number is stored as an array of limbs in a base such as 2^32 or 10^9, often with little-endian ordering. Operations are built from low-level digitwise algorithms and higher-level arithmetic, ensuring correct carry handling and sign management. For very large values, specialized algorithms and data structures, such as FFT-based multiplication or modular reduction, improve performance.

Supported operations include addition, subtraction, multiplication, division, modular arithmetic, and exponentiation. Algorithms evolve to balance speed and memory usage, with techniques like Karatsuba, Toom-Cook, and fast Fourier transform methods used for large operands. Common tasks also involve primality testing, gcd computations, and modular inversion, which underpin many cryptographic protocols.

Common implementations span multiple ecosystems. Languages such as Python provide built-in arbitrary-precision integers; Java offers BigInteger; C/C++ often relies on libraries like GNU MP or OpenSSL BIGNUM. In some environments, a type named LARGE_INTEGER denotes a fixed 64-bit integer, used for system APIs and binary interfaces.

Applications of large integers include cryptography (RSA, ECC), computational number theory, and simulations requiring exact arithmetic beyond standard word sizes.