Kunerth's algorithm

Kunerth's algorithm is an algorithm for computing the modular square root of a given number. The algorithm does not require the factorization of the modulus, and uses modular operations that are often easy when the given number is prime.

Algorithm

To find $y$ from a given value

$B=y^{2}modN,$

it takes the following steps:

1. Find the modular square root $r\equiv \sqrt{\pm N}(\mathrm{mod}B)$. This step is quite easy when $B$ is a prime, irrespective of how large $N$ is.
2. Solve a quadratic equation associated with the modular square root of $w^2=A\cdot z^2+B\cdot z+C$. Most of Kunerth's examples in his original paper solve this equation by having $C$ be a integer square and thus setting $z$ to zero.

   Expand the left hand side of the following equation:

   $((B\cdot z+r{)}^2+(B\cdot F\mp N))/B=w^2.$

   Expanding the left hand side results in a quadratic form $A\cdot z^2+D\cdot z+C+F$. One can then make sure that the equation can be solved by adjusting $F$ so as to make $C+F$ a square.

3. Having solved the associated quadratic equation we now have the variables $w$ and set $v$ = $r$ (if $C$ in the quadratic is a natural square).
4. Solve for variables $\alpha$ and $\beta$ the following equation:

   $\alpha =w(v+w\beta ),$

5. Obtain a value for $X$ via factorization of the following polynomial:

   ${\alpha }^2\cdot x^2+(2\alpha \beta -N)x+({\beta }^2-(y^{2}modN))$

   obtaining an answer like

   $(-37+9x)(1+25x)$

6. Obtain the modular square root by the equation. Remember to set $X$ such that the term above is zero. Thus $X$ would be 37/9 or -1/25.

   $y\equiv \alpha X+\beta (\mathrm{mod}N).$

Example

To obtain $\sqrt{41}mod856,$ first obtain $\sqrt{-856}\equiv 13(\mathrm{mod}41)$.

Then expand the polynomial:

$((41z+13{)}^2+856)/41=w^2$

into

$25+26z+41z^2$

Since, in this case the C term is a square, we take $w=5$ and compute $v=13$ (in general, $v=41\cdot z+13$).

Solve for $\alpha$ and $\beta$ the following equation

$\alpha ==w(v+w\beta )$

getting the solution $\alpha =15$ and $\beta =-2$. (There may be other pairs of solutions to this equation.)

Then factor the following polynomial:

${\alpha }^2{x}^2+(2\alpha \beta -856)x+({\beta }^2-41)$

obtaining

$(-37+9x)(1+25x)$

Then obtain the modular square root via

$15\cdot (37\cdot 9^{-1})+(-2)\equiv 345(\mathrm{mod}856).$

Verify that $345^2\equiv 41(\mathrm{mod}856).$

In the case that $\sqrt{-856}mod41$ has no answer, then $r\equiv \sqrt{-856}(\mathrm{mod}b\cdot 856+41)$ can be used instead.

See also

• Methods of computing square roots

References

• Adolf Kunerth, "Sitzungsberichte. Academie Der Wissenschaften" vol 75, II, 1877, pp. 7–58
• Adolf Kunerth, "Sitzungsberichte. Academie Der Wissenschaften" vol 82, II, 1880, pp. 342–375