Pocklington's algorithm

Pocklington's algorithm is a technique for solving a congruence of the form

where x and a are integers and a is a quadratic residue.

The algorithm is one of the first efficient methods to solve such a congruence. It was described by H.C. Pocklington in 1917.

The algorithm

(Note: all are taken to mean , unless indicated otherwise.)

Inputs:

• p, an odd prime
• a, an integer which is a quadratic residue .

Outputs:

• x, an integer satisfying . Note that if x is a solution, −x is a solution as well and since p is odd, . So there is always a second solution when one is found.

Solution method

Pocklington separates 3 different cases for p:

The first case, if , with , the solution is .

The second case, if , with and

1. , the solution is .
2. , 2 is a (quadratic) non-residue so . This means that so is a solution of . Hence or, if y is odd, .

The third case, if , put , so the equation to solve becomes . Now find by trial and error and so that is a quadratic non-residue. Furthermore, let

.

The following equalities now hold:

.

Supposing that p is of the form (which is true if p is of the form ), D is a quadratic residue and . Now the equations

give a solution .

Let . Then . This means that either or is divisible by p. If it is , put and proceed similarly with . Not every is divisible by p, for is not. The case with m odd is impossible, because holds and this would mean that is congruent to a quadratic non-residue, which is a contradiction. So this loop stops when for a particular l. This gives , and because is a quadratic residue, l must be even. Put . Then . So the solution of is got by solving the linear congruence .

Examples

The following are 4 examples, corresponding to the 3 different cases in which Pocklington divided forms of p. All are taken with the modulus in the example.

Example 0

This is the first case, according to the algorithm, , but then not 43, so we should not apply the algorithm at all. The reason why the algorithm is not applicable is that a=43 is a quadratic non residue for p=47.

Example 1

Solve the congruence

The modulus is 23. This is , so . The solution should be , which is indeed true: .

Example 2

Solve the congruence

The modulus is 13. This is , so . Now verifying . So the solution is . This is indeed true: .

Example 3

Solve the congruence . For this, write . First find a and such that is a quadratic nonresidue. Take for example . Now find , by computing

And similarly such that

Since , the equation which leads to solving the equation . This has solution . Indeed, .

References

• Leonard Eugene Dickson, "History Of The Theory Of Numbers" vol 1 p 222, Chelsea Publishing 1952