Can secp256k1 have different parameters and still be called secp256k1?

Question

The secp256k1 standard (as defined in https://www.secg.org/sec2-v2.pdf, section 2.4.1) has a set of recommended parameters.

• Is there a particular reason why these parameters are just 'recommended', and not 'required'?
• Are we still talking about 'secp256k1' if different parameters are used?

Answer 1

The SEC2 document recommends the use of a number of curves. One of these curves is called secp256k1.

The recommendation doesn't refer to the parameters. What is recommended is the use of the secp256k1 curve and the parameters are integral to how the curve is defined. If you use any other parameters, you can't call it secp256k1.