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ECDHE is quite widely used and recommend. And it works with elliptical keys and provides forward secrecy. It's used for the key exchange.

ECDSA is not widely used though, but it does also use elliptical keys. It it used for authentication the server via (TLS) certificates.

More information about the differences here: ECDSA vs ECDH vs Ed25519 vs Curve25519

  • However regarding the usage in browsers is there any difference between the curves used in ECDHE and ECDSA?
  • Do browsers support the same curves for every elliptical curves algorithm? Do they possibly even always used the same curve for ECDHE and ECDSA in TLS_ECDHE_ECDSA?
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However regarding the usage in browsers is there any difference between the curves used in ECDHE and ECDSA?

No, there probably is not. As the key establishment and certificate verification is likely to be in the same crypto API it is pretty likely that the same curves are supported (or not).

Do browsers support the same curves for every elliptical curves algorithm?

That's pretty likely because of the same reasons.

Do they possibly even always used the same curve for ECDHE and ECDSA in TLS_ECDHE_ECDSA?

The curve for the ECDSA part depends on the certificate used. The key in the certificate is static while the ECDH keys are ephemeral. So it is likely that shorter curves are used for ECDH than ECDSA. There is not a direct connection between the two in the protocol, so although the same curve may be used, it is certainly not a given.

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