# Clause 9 Elliptic curves, added in the third edition

The elliptic curve family entered the 2009 edition and is the newest arithmetic in the part. It gives short keys and a different coupon profile, with the verifier able to precompute in one subclause and no coupon at all in another. The fourth edition carries the family forward.

## What it asks

Record the curve, the base point, the subgroup order and the coupon arrangement for each deployment, and show that the curve was selected from an approved set rather than generated locally. Test that a point not on the curve and a point outside the subgroup are both rejected.

## What the revision does

The fourth edition keeps the family and will restate it. The public record stops before the revised text.

## Who it binds

The cryptography owner and the implementer of the point validation.

## Evidence an auditor samples

- The curve and base point record per deployment (cryptography standard)
- The curve selection review (design note)
- The point validation tests, on curve and in subgroup (test suite)

## The test you can run

Present a point that is not on the curve. If the verifier proceeds, validation is missing.

## Common failures

- The curve is generated locally with unexplained parameters.
- Validation checks the coordinates but not the subgroup.

## How to close it

Use an approved curve and test both validations. Curve parameters are hard to check by eye, so the selection record and the validation tests are the only practical evidence.

## Source

ISO/IEC 9798-5:2009(E), Clause 9, 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.