Why the Check Value, Not the Radio, Stops MITM

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

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Ada ADA · CALCULATION AUDIT

Why the Check Value, Not the Radio, Stops MITM

At the calculator’s default sliders — 10 attempts against 100 devices — Just Works surrenders essentially the whole fleet (about 100 devices, a 99.99990% per-device compromise), while Numeric Comparison’s 6-digit check leaves it at roughly 0.001 devices. The same attacker and the same attempts are separated by a factor near 100,000. This audit rebuilds both sides and asks whether that gap comes from the radio, the range, or the encryption — or entirely from the six-digit human check.

Companion to the chapter BLE Pairing Practice Checks — every number here comes from that chapter.

Ada: The calculator above claims that over the same handful of attempts, Just Works exposes almost every device while Numeric Comparison exposes almost none. That gap is the entire argument for adding a check value, so let me reproduce it from the model’s own illustrative rates at its default sliders: 10 attempts, 100 devices.

Just Works treats each of the 10 setup attempts as an independent timing race the attacker wins with the model’s illustrative probability 0.75:

  • Per-device escape from capture over 10 tries: (1 - 0.75)^10 = 0.25^10 = 0.0000009536743
  • Per-device compromise probability: 1 - 0.0000009536743 = 0.9999990463, i.e. 99.99990%
  • Expected devices compromised in the fleet: 100 x 0.9999990463 = 99.99990, which the panel rounds to 100 devices

Numeric Comparison instead forces the attacker to guess a 6-digit value, so each attempt succeeds with probability 1 / 1,000,000:

  • Per-device compromise over 10 tries: 1 - (1 - 0.000001)^10 = 0.000009999955, i.e. about 0.001%
  • Expected devices compromised: 100 x 0.000009999955 = 0.0009999955, about 0.001 devices

Dividing one by the other, 0.9999990463 / 0.000009999955 = 100,000.4: the same attacker, the same 10 attempts, and the same 100 devices, yet Just Works is roughly a hundred thousand times more likely to surrender a device than Numeric Comparison. The defence in that ratio comes entirely from the six-digit human check, not from the radio, the range, or the encryption, which is exactly why a displayless lock that can only fall back to Just Works needs an out-of-band channel or a physical setup action before it ships.

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