I am trying to determine the multiplicative inverse of $47$ modulo $64$. So I have looked for an algorithm or scheme in order to perform this.

I found this wiki explaining how to find a multiplicative inverse. I tried to perform all the calculations, but the result was incorrect. I got $5$ as a multiplicative inverse, but this cannot be true: $47\times5\not\equiv1\pmod{64}$. Who can help me?

  • $\begingroup$ Take a look at section 6 in ia.cr/2017/411 $\endgroup$
    – j.p.
    May 18 '17 at 7:11

A boring method is to carefully apply the (partially) extended Euclidean algorithm.

But in the question, the modulus is a power of two (specifically $2^6$), and we can use that $$a\,x\equiv1\pmod{2^k}\implies a\,x\,(2-a\,x)\equiv1\pmod{2^{2k}}$$ from which it follows this fact:

if the modular inverse of $a$ modulo $2^k$ is (the lower $k$ bits of) $x$, then
the modular inverse of $a$ modulo $2^{2k}$ is (the lower $2k$ bits of) $x\,(2-a\,x)$
(where negative integers are in 2's-complement convention, dominant in modern CPUs).

This not-so-much-known fact allows computation of multiplicative inverse modulo $2^k$. We start from an inverse $x$ of $a$ over few bits (that can be $x=a$, perhaps $\bmod 7$, which is the inverse for any odd $a$ over three bits), and iterate $x\gets x\,(2-a\,x)$, possibly truncated to the number of known-correct result bits. That number of bits doubles at each iteration, thus about $\log_2(k)$ steps are enough, and it is only used product, subtraction, and bit truncation on values no wider than $k$ bits. That is blindingly fast compared to the Euclidean algorithm's $O(k)$ steps; and eases getting data-independent execution time, which comes handy in some cryptographic computations (e.g. the preliminary computation of $m'$ in Montgomery multiplication, algorithm 14.36 of Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone's Handbook of Applied Cryptography).

I learned the technique from Colin Plumb's Computing multiplicative inverses (post on sci.crypt with Message-ID: <1994Apr6.093116.27805@mnemosyne.cs.du.edu>, 1994). His statement applies to inverse modulo a prime power, and points the relation to the Newton's iteration for finding $x = 1/a$ in $\Bbb R$.

A modern exposition, with benchmarks, is in Jean-Guillaume Dumas: On Newton-Raphson iteration for multiplicative inverses modulo prime powers.

A bibliography, and other techniques faster than the Euclidean algorithm, are in Çetin Kaya Koç: A New Algorithm for Inversion mod $p^k$.

Here, to perform the desired computation quickly, we use $k=3$, $a=47$, and compute $a\bmod2^k=47\bmod8=7$, which multiplicative inverse modulo $8$ is also $x=7$. Now we compute $$\begin{align} (x\,(2-a\,x))\bmod2^{2k}&=(7\,(2-47\times7))\bmod 64\\ &=15\end{align}$$

Hence the desired modular inverse of $47$ modulo $64$ is $15$.

  • $\begingroup$ 7 (2 -47x7) is equal to 2289 ? $\endgroup$ May 24 '17 at 9:02
  • $\begingroup$ @user3834282: Actually, $7(2-47\times 7)$ is $-2289$, and $(-2289)\bmod64$ is $15$. Recall the definition of $\bmod$ as an operator: $u\bmod v$ is the (uniquely defined) $w$ such that $0\le w<v$ and $v$ divides $u-w$. For $u\ge0$, $w$ is the remainder of the Euclidean division of $u$ by $v$. For negative $u$, one can use the trivially established: $(u\bmod v)=(v-1)-((1-u)\bmod v)$. Apply with $u=-2289$ and $v=64$. Do not confuse with $w\equiv u\pmod v $ [notice the $\equiv$ and the $($ immediately before $\bmod$], which only tells that $v$ divides $u-w$, and does not uniquely define $w$. $\endgroup$
    – fgrieu
    May 24 '17 at 9:28

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