## Cryptology ePrint Archive: Report 2017/411

A New Algorithm for Inversion mod $p^k$

Çetin Kaya Koç

Abstract: A new algorithm for computing $x=a^{-1}\pmod{p^k}$ is introduced. It is based on the exact solution of linear equations using $p$-adic expansions. It starts with the initial value $c=a^{-1}\pmod{p}$ and iteratively computes the digits of the inverse $x=a^{-1}\pmod{p^k}$ in base $p$. The mod 2 version of the algorithm is significantly more efficient than the existing algorithms for small values of $k$. We also describe and analyze all existing algorithms, and compare them to the proposed algorithm. Our algorithm stands out as being the only one that works for any $p$, any $k$, and digit-by-digit. Moreover it requires the minimal number of arithmetic operations (just a single addition) per step.

Category / Keywords: number theory, arithmetic, cryptography

Date: received 11 May 2017, last revised 28 Jun 2017

Contact author: koc at cs ucsb edu

Available format(s): PDF | BibTeX Citation

Note: Small improvements in the paper. This is the final version.

Short URL: ia.cr/2017/411

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