sqrtmk

Sqrtmk is a hypothetical algorithm and accompanying software library described in theoretical discussions of square-root extraction in modular arithmetic and fixed-precision computation. The name combines sqrt, indicating a square root operation, with MK as a placeholder for a mathematical construct or modulus in which roots are sought. It is not a standard or widely adopted method, but rather a conceptual framework used for exploration and education.

In discussions, sqrtmk is presented as a two-stage approach to modular square roots. The first stage uses established prime-modulus techniques, such as the Tonelli–Shanks method, to find a root when the modulus is prime or to determine residue status. The second stage extends the result to more complex moduli via lifting methods (for example, Hensel lifting) or the Chinese Remainder Theorem, aiming to produce roots in composite or structured rings. To improve performance, the method often invokes Montgomery reduction or other fast modular arithmetic representations to replace division with multiplications.

Implementation ideas for sqrtmk focus on modular arithmetic libraries capable of handling arbitrary-precision integers. A typical API would accept an integer n and a modulus m, returning a square root of n modulo m when it exists, or signaling non-residue status. Verification commonly involves squaring the result and reducing modulo m to confirm equality with n.

Applications discussed in theory include cryptographic research, primality testing contexts, and educational demonstrations of how different lifting and reduction techniques interact. Limitations emphasize that sqrtmk is not standardized and may depend on the factorization of the modulus, with not all moduli admitting a square root. See also Tonelli–Shanks algorithm, Chinese Remainder Theorem, Hensel lifting, and Montgomery reduction.