Here is a game you can play. It is essentially the simplified version of poker over the telephone from Section 18.2. There are five cards: ten, jack, queen, king, ace. They are shuffled and disguised by raising their numbers to a random exponent mod the prime 24691313099. You are supposed to guess which one is the ace.

To start, pick a random exponent. We use the colon after khide() so that we cannot cheat and see what value of $k$ is being used.

> k:= khide():

Now, shuffle the disguised cards (their numbers are raised to the $k$th power mod $p$ and then randomly permuted):

> shuffle(k)

   [14001090567, 16098641856, 23340023892, 20919427041, 7768690848]

These are the five cards. None looks like the ace that’s because their numbers have been raised to powers mod the prime. Make a guess anyway. Let’s see if you’re correct.

> reveal(%)

             ["ten", "ace", "queen", "jack", "king"]

Let’s play again:

> k:= khide():

> shuffle(k)

   [13015921305, 14788966861, 23855418969, 22566749952, 8361552666]

Make your guess (note that the numbers are different because a different random exponent was used). Were you lucky?

> reveal(%)

                 ["ten", "queen", "ace", "king", "jack"]

Perhaps you need some help. Let’s play one more time:

> k:= khide():
> shuffle(k)

   [13471751030, 20108480083, 8636729758, 14735216549, 11884022059]

We now ask for advice:

> advise(%)

                3

We are advised that the third card is the ace. Let’s see (recall that %% is used to refer to the next to last output):

> reveal(%%)

               ["jack", "ten", "ace", "queen", "king"]

How does this work? Read the part on “How to Cheat” in Section 18.2. Note that if we raise the numbers for the cards to the $(p-1)/2$ power mod $p\text{,}$ we get

> map(x->x&$\stackrel{\mathbf{ˆ}}{\phantom{\mathbf{a}}}$((24691313099-1)/2) mod 24691313099,
[200514, 10010311, 1721050514, 11091407, 10305])

                     [1, 1, 1, 1, 24691313098]

Therefore, only the ace is a quadratic nonresidue mod $p\text{.}$