Can Humpty Dumpty be Put Back?

Here’s something that every one of us knows from childhood:

-         It is the easiest thing to multiply two numbers, no matter how large they are (It may take time, but the “how” part is child’s play. Literally.);

-         But there is no easy way to find the factors of a number, other than hit and trial, which can take forever.

Such operations (easy in one direction, but impossibly difficult in the other) are called “Humpty Dumpty functions”, after the most famous egg in history.

This belief in the Humpty Dumpty nature of the problem of factorization is at the heart of most encryption algorithms, including the ones used for your financial transactions on the Internet. No mathematician has found a way to factorize a number easily; so the belief is well placed indeed.

Or is it?

In his book, The Man who Mistook his Wife for a Hat, Oliver Sacks noticed something very surprising about a pair of twins who were considered as autistics, retards or idiot savants:

“When one comes to their ability to calculate-the typical forte of arithmetical prodigies and ‘mental calculator’- they do astonishingly badly, as badly as their IQ’s of sixty might lead one to think. They cannot do simple addition or subtraction with any accuracy, and cannot even comprehend what multiplication or division means.”

And yet:

“When a box of matches fell to the ground, one said 111 and the other murmured 37 thrice. Their point? The number of matches that had fallen was 111; and why 37?

They said in unison, ’37, 37, 37, 111.’”

That led Sacks to wonder:

“They had then gone to ‘factor’ the number 111- without having any method, without even ‘knowing’ (in the ordinary way) what factors meant.”

Does this mean that there is a different way to factorize a number that doesn’t occur to “normal” people like us? Do the twins prove that the factorization problem is not a Humpty Dumpty function? After all, as Nick Bostrom wrote in his book, Superintelligence:

“Minor changes in brain volume and wiring can have major consequences.”

Retards and autistics do appear to be wired differently, don’t they?

If this is true, what does it say about the security of our encryption algorithms? Could it mean the end of e-commerce, until the next algorithm is devised?