Math Bridge: Wearable Security and Battery Trade-Off

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Math BridgeApplicationsStruggle-friendly runway

Which costs more charge: encryption or shelf time?

Put hardware, security, and product timing on one battery ledger.

Battery Bruno, the energy guideBattery Bruno guides
The one targetCompare security overhead and self-discharge on the same 180 mAh pack.
The chapter case3.7 V LiPo, 15% AES, 7.5% ChaCha20, 2.5% monthly shelf loss.
What it buys youA cross-team trade-off stated in common units and elapsed time.

A field team faces an unresolved physical question: Which costs more charge: encryption or shelf time? They must answer it before changing shelf time on the real device. Predict the direction first.

See the relationship before changing it

The figure reads from left to right. The blue card is shelf time. The middle card applies this page's relationship. The green card is nameplate energy. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This added model holds every other chapter fixture fixed, so the numeric fixture does not switch without explanation.

Shelf time changes nameplate energy An input card leads through the page relationship to the nameplate energy result. SET INPUT ONE CONTROL APPLY RULE predict calculate check units READ RESULT
Walk the arrows. The comparison does not say to weaken security. It shows hardware, firmware, and product teams which measured terms belong in the same decision record.

Derive the baseline in four named moves

  1. 1

    Name the input. The chapter baseline for shelf time is 3.

  2. 2

    Name the relationship. E=(180/1000)(3.7)=0.666 Wh AES: 180(1-0.15)=153 mAh; 153/(180/7)=5.95 days ChaCha20: 180(1-0.075)=166.5 mAh; 166.5/(180/7)=6.475 days shelf loss=100[1-(0.975)³]=7.314% Vsag=(0.120)(0.15 to 0.45)=18 to 54 mV

  3. 3

    Substitute the chapter fixture. Set shelf time to 3. The page ledger gives nameplate energy as 0.666 Wh.

  4. 4

    Read the result. Keep Wh beside the value. Use it only inside the technical boundary on this page.

Predict, then change shelf time

Try Predict the direction of nameplate energy. Move one control, calculate, then check your prediction.

3
Chapter baseline
Nameplate energy

Observe The comparison does not say to weaken security. It shows hardware, firmware, and product teams which measured terms belong in the same decision record. Reset the control to 3 and compare nameplate energy.

Explain Only shelf time moves here. The other chapter fixtures remain fixed.

Check yourself

What should you do before trusting a moved-control result?
Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.
What does this small model leave out?
Answer: Only shelf time moves. Field effects named in the page's technical boundary stay fixed.

Honesty note. This teaching model changes only shelf time. It replays the chapter fixture; it is not a field prediction.

1. Start with the physical story

A security choice spends charge during use. A product also loses charge while it waits on a shelf. Both losses draw from the same pack, so comparing percentages only works when their time bases and starting capacity are explicit.

Battery Bruno: One percentage belongs to a workload; the other belongs to a calendar. Put both on the same ledger before debating them.

2. Name every algebra move

1

Convert energyDivide 180 mAh by 1,000 and multiply by 3.7 V.

2

Find daily chargeDivide 180 mAh by the seven-day baseline.

3

Apply AES overheadMultiply capacity by one minus 0.15.

4

Apply ChaCha overheadMultiply capacity by one minus 0.075.

5

Compound shelf retentionRaise 0.975 to the number of months.

6

Check pulse sagMultiply the 0.120 A burst by fresh and aged resistance.

3. Reproduce the chapter case

E=(180/1000)(3.7)=0.666 Wh
AES: 180(1−0.15)=153 mAh; 153/(180/7)=5.95 days
ChaCha20: 180(1−0.075)=166.5 mAh; 166.5/(180/7)=6.475 days
shelf loss=100[1−(0.975)³]=7.314%
Vsag=(0.120)(0.15 to 0.45)=18 to 54 mV

At three months, shelf loss is almost the chapter's entire 7.5% ChaCha20 overhead, while pulse sag remains small in this illustrative pack.

4. Try one real input

TryMove shelf time and compare calendar loss with the fixed security paths.

Shelf time
Nameplate energy
AES charge
AES runtime
ChaCha20 charge
ChaCha20 runtime
Charge after shelf time
Shelf loss
ChaCha20 runtime after shelf time
Fresh burst sag
Aged burst sag

ObserveSecurity-path numbers stay fixed while shelf loss compounds. After several months, product timing can dominate the difference between the two algorithms.

ExplainThe comparison does not say to weaken security. It shows hardware, firmware, and product teams which measured terms belong in the same decision record.

Technical boundaries.

The overhead percentages are the chapter's scenario inputs, not universal algorithm measurements.

Security
Energy depends on processor, implementation, key setup, message size, radio retries, protocol, and threat controls.
Battery
Self-discharge and resistance vary with chemistry, temperature, state of charge, age, and cell quality.
Product
Shelf time, shipping state, activation, update cadence, and service policy belong in the release decision.

Correct, not complete: this ledger does not rank cryptographic safety or approve a wearable runtime claim.

5. Use the result in the design

Profile the real security path on target hardware, measure cell loss over the actual fulfilment interval, and keep security requirements fixed while reducing avoidable wake time and retries.

6. Record the evidence state

Keep algorithm and implementation, hardware, payload, transaction count, current trace, battery chemistry and lot, manufacture and activation dates, temperature, shelf state, measured capacity, and threat requirement.

7. Check yourself

Why is 0.666 Wh different from 180 mAh?
Answer: Watt-hours include voltage and represent energy; milliamp-hours represent charge.
Why does three-month loss compound?
Answer: Each month retains 97.5% of what remained, so retention is 0.975³ rather than 1−3×0.025 as an exact model.
Does lower energy overhead make an algorithm acceptable?
Answer: No. Required security properties, implementation quality, and measured system behavior come first.
Honesty boundary.

The arithmetic reproduces the chapter's 180 mAh, AES-256, ChaCha20, and three-month shelf example.

Security
Energy depends on processor, implementation, key setup, message size, radio retries, protocol, and threat controls.
Battery
Self-discharge and resistance vary with chemistry, temperature, state of charge, age, and cell quality.
Product
Shelf time, shipping state, activation, update cadence, and service policy belong in the release decision.

Correct, not complete: this ledger does not rank cryptographic safety or approve a wearable runtime claim.

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