In this paper Simplifying Quotient Determination in High-Radix Modular Multiplication, the authors have proposed to replace the original modulus $M$ in Montgomery Multiplication $ABR^{-1} \bmod{M}$ to a new value $\tilde{M} = (M' \bmod{2^k}) M$, where $(-MM')\bmod{2^k}=1$.

The new modulus $\tilde{M}$ is used to improve the original Montgomery multiplication algorithm but also gives an output in the range of $[0, 2\tilde{M})$ instead of the range $[0, 2M)$ I actually need. However, the paper does not mention how to convert the result back to the desired range. What should I do to retrieve the actual result of $ABR^{-1} \bmod{M}$?

  • $\begingroup$ taking $\mod M$ doesn't enough? $\endgroup$ – kelalaka Feb 10 at 20:22

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