A Constant-Time Approximate Multiplication Algorithm for Arbitrarily Large Integers.

I present a simple yet powerful method for approximating the multiplication of extremely large integers numbers with hundreds of thousands or millions of digits by combining power‑of‑two decomposition with controlled mantissa truncation. Unlike classical big‑integer algorithms, which scale with input size and rely on exact arithmetic, this approach preserves only the most significant bits of each operand, producing a result whose relative error becomes microscopic as the numbers grow. Empirically, the approximation achieves speedups of 80× to over 1000× compared to GMP, while maintaining relative errors on the order of 10⁻³⁰⁸, effectively indistinguishable from the exact product. When the truncation threshold exceeds the mantissa size, the method seamlessly transitions into exact multiplication, matching GMP both in accuracy and performance. This hybrid behavior—exact for moderate sizes, constant‑time and highly accurate for enormous sizes—reveals an unexplored numerical regime where big‑integer multiplication can be approximated with extraordinary fidelity. To the best of our knowledge, we are not aware of previous work describing this particular approximation strategy for arbitrary-precision integer multiplication based on power-of-two decomposition and truncation of the odd component., offering a new perspective that bridges numerical analysis and integer arithmetic for unprecedentedly large inputs. For fixed truncation precision t, the algorithm requires O(t) working memory, independent of the operand length (excluding storage or streaming of the input operands).