# 7.3 The exchange with a composite modulus

The exchange follows the same commitment, challenge and response shape, with the composite arithmetic behind it. The verifier has to check the response against the right base and modulus, and a mismatch between the two is silent unless the test proves otherwise.

## What it asks

Record the base and modulus the verifier uses, and show they match the ones the claimant used. Test a response computed under a different base, a replayed challenge and a wrong key, and keep the outputs with the release record.

## What the revision does

The fourth edition will restate the exchange. No public copy of the new sequence exists yet.

## Who it binds

The implementer and the tester.

## Evidence an auditor samples

- The exchange sequence with the base and modulus named (interface specification)
- The verifier configuration that pins both (configuration management database)
- The three negative tests and their outputs (test report)

## The test you can run

Change the base in the verifier configuration and run the positive test. If it still passes, the base is not pinned.

## Common failures

- The verifier reads the base from the claimant message.
- The modulus is shared but the base is per deployment, with no record.

## How to close it

Pin both values and test the pin by changing one. A verifier that accepts the claimant arithmetic context has no way to know which mechanism it is running.

## Source

ISO/IEC 9798-5:2009(E), 7.3, publisher preview; replaced by ISO/IEC DIS 9798-5, stage 40.20. Read on 21 October 2026. Our wording, not the instrument's.
https://www.iso.org/standard/89516.html

Not legal advice; does not certify.