No, ~~*R*~~ *k* (edit: see below) is a random value, which must never have been used to sign any other message (or else the private key can be solved for algebraically). In some schemes, ~~*R*~~ *k* is deterministically derived from the message to ensure the preceding requirement is satisfied.

They are using P for two (slightly) different things there which is confusing.

In the top part P is the random^1 curve point selected by the signer.

In the bottom part P is the curve point computed by the verifier.

For a correct signature these should be the same but it would be clearer to use a different variable name (e.g. P’) for the second one, because the point of doing the verification is to see if the right result is obtained.

Anyway to answer your question, P and R are related but not exactly the same thing, as written there R is the x-coordinate of P. IOW R is a number while P is a point (on the elliptic curve).

What the verifier does to check the signature is to compute P’ and see if it has x-coordinate equal to R. If so then the signature is valid.

^1 (edit): as u/whitslack pointed out, it is not necessarily random but the corresponding k does need to be secret and also not reused. https://www.rfc-editor.org/rfc/rfc6979.html#section-3.2 is one specification for computing k deterministically (P = k*G)

*if* S is correct*

Sir this is a wendys.

PV=nRT flashback to chemistry….

No, ~~*R*~~ *k* (edit: see below) is a random value, which must never have been used to sign any other message (or else the private key can be solved for algebraically). In some schemes, ~~*R*~~ *k* is deterministically derived from the message to ensure the preceding requirement is satisfied.

They are using P for two (slightly) different things there which is confusing.

In the top part P is the random^1 curve point selected by the signer.

In the bottom part P is the curve point computed by the verifier.

For a correct signature these should be the same but it would be clearer to use a different variable name (e.g. P’) for the second one, because the point of doing the verification is to see if the right result is obtained.

Anyway to answer your question, P and R are related but not exactly the same thing, as written there R is the x-coordinate of P. IOW R is a number while P is a point (on the elliptic curve).

What the verifier does to check the signature is to compute P’ and see if it has x-coordinate equal to R. If so then the signature is valid.

^1 (edit): as u/whitslack pointed out, it is not necessarily random but the corresponding k does need to be secret and also not reused. https://www.rfc-editor.org/rfc/rfc6979.html#section-3.2 is one specification for computing k deterministically (P = k*G)