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Start by explaining the need for a safe modular multiplication to avoid overflow when multiplying two numbers up to 2^61-1. Then present binary exponentiation (exponentiation by squaring) that uses this helper. Finally, discuss tests and complexity.
Pro tip: Mention that using unsigned 64-bit integers and the Russian peasant multiplication (double-and-add) with conditional subtraction ensures no overflow, and highlight that this approach is constant-time with respect to the exponent, which is important for cryptographic applications.
Restate that a, b, m can be up to 2^61-1, so direct multiplication overflows 64-bit. Emphasize the need for a safe modular multiplication helper.
Implement a function that computes (a * b) % m without overflow using either Russian peasant multiplication (double-and-add) or splitting into high/low 32-bit parts. Ensure it handles m=1 and negative inputs if applicable.
Use binary exponentiation (exponentiation by squaring) with the safe multiplication helper. Handle edge cases: b=0 (result 1 % m), m=1 (result 0), and a >= m (reduce a modulo m first).
Include tests for small values (compare with naive), edge cases (b=0, m=1, a=0), and large values near 2^61-1 to verify no overflow and correctness.
State that modular multiplication is O(log b) and exponentiation is O(log b) multiplications, so overall O(log b) time and O(1) space.
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