What Rate Limiting Does to a 6-Digit Passkey

Ada re-derives this chapter’s own numbers step by step, at full precision

ADA · CALCULATION AUDIT

What Rate Limiting Does to a 6-Digit Passkey

A BLE device protects pairing with a random 6-digit passkey, one of 1,000,000 values, and the chapter clocks a brute-force attacker at 13.89 hours when nothing throttles 10 guesses a second. Add a retry delay of one attempt per 30 s and the same search stretches to 174 days — a 300x slowdown. This audit rebuilds both times and asks whether that 300x comes from the length of the passkey or purely from how hard the device throttles retries.

Companion to the chapter Encryption and the Key Hierarchy — every number here comes from that chapter.

Ada: The Putting-Numbers panel gives a passkey brute-force time with and without rate limiting. Let me rebuild both from T_attack = (2^entropy / R) x 0.5 and see exactly where the protection comes from.

A random 6-digit passkey has log2(1,000,000) = 19.9316 bits of entropy, so 2^19.9316 = 1,000,000 guesses:

• Unlimited, 10 attempts/second: T = (1,000,000 / 10) x 0.5 = 50,000 s, and 50,000 / 3,600 = 13.89 hours.
• Rate-limited to one attempt per 30 s (R = 1/30): T = (1,000,000 / (1/30)) x 0.5 = 1,000,000 x 30 x 0.5 = 15,000,000 s, and 15,000,000 / 86,400 = 173.6 days, about the stated 174 days.

The design meaning is in the ratio: 15,000,000 / 50,000 = 300x, and that 300 is just the rate ratio 10 / (1/30) = 300 – the entropy term cancels entirely. Rate limiting’s slowdown is set purely by how hard you throttle retries, not by how long the PIN is, which is why a modest 20-bit passkey plus a retry delay turns a same-afternoon attack (about 14 hours) into a months-long one (about 174 days).

Every number above is taken from the chapter’s own material and re-derived step by step.