The UK's GCHQ, home to much of the country's surveillance operations.
Photo: GCHQ / MOD/EPA

The politicians calling for a "backdoor key" don't know what they're talking about. 

By Philip Elmer-DeWitt
December 22, 2015

It’s only Tuesday and already this week Apple has come out twice—guns a-blazing—against efforts by law enforcement agencies to weaken public-key encryption in the name of national security.

The issue came up in the 60 Minutes interview with Apple CEO Tim Cook that aired Sunday, and it surfaced again Monday when the company filed an eight-page brief opposing Britain’s Investigatory Powers Bill—the so-called “snooper charter.”

The political winds may shift—for encryption when the NSA runs amok, against encryption when terrorists strike—but the crux of the matter never changes. It’s a matter of arithmetic.

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The only way we know how to protect privacy, Cook told 60 Minutes‘ Charlie Rose, is with encryption.

Modern encryption, as former Apple executive Jean Louis Gassée explained last week in Let’s Outlaw Math, is based on a simple mathematical fact. It’s easy to calculate the product of two prime numbers, but going the other way—breaking a long number into its prime factors—gets exponentially harder the longer that number gets.

Public key encryption—the kind that ships in devices made by Apple AAPL and Google AAPL —is based on the difficulty of solving that math problem.

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In one 2009 experiment, it took hundreds of computers two years to guess the prime factor of a single 232-digit number. Researchers estimated that a 1024-bit key would take 1,000 times longer.

The “backdoor key” police say they need to fight crime would have to somehow cut through thousands of years of number crunching without defeating that purpose. As Apple put it in its Monday submission to the British Parliament:

“A key left under the doormat would not just be there for the good guys. The bad guys would find it too.”

When politicians say we need backdoor keys for own protection, writes Gassée, “one is tempted to ask, glibly, if these leaders are ignorant, delusional, or dishonest—or all of the above.”

Follow Philip Elmer-DeWitt on Twitter at @philiped. Read his Apple coverage at fortune.com/ped or subscribe via his RSS feed.

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