JWT, how to verify signature?

I have this JWT:

eyJhbGciOiJSUzI1NiIsImtpZCI6IjVkMzQwZGRiYzNjNWJhY2M0Y2VlMWZiOWQxNmU5ODM3ZWM2MTYzZWIiLCJ0eXAiOiJKV1QifQ.eyJuYW1lIjoiemFnYWxvIiwiaXNzIjoiaHR0cHM6Ly9zZWN1cmV0b2tlbi5nb29nbGUuY29tL3Byb2ZlcHQtM2M0NzkiLCJhdWQiOiJwcm9mZXB0LTNjNDc5IiwiYXV0aF90aW1lIjoxNjY2MjkxNDAzLCJ1c2VyX2lkIjoiZ1JtdnFYb0tySE85T0RLUURCYTBWNnRaNTBLMiIsInN1YiI6ImdSbXZxWG9LckhPOU9ES1FEQmEwVjZ0WjUwSzIiLCJpYXQiOjE2NjYyOTE0MDMsImV4cCI6MTY2NjI5NTAwMywiZW1haWwiOiJyc2p1bGlhb0BnbWFpbC5jb20iLCJlbWFpbF92ZXJpZmllZCI6ZmFsc2UsImZpcmViYXNlIjp7ImlkZW50aXRpZXMiOnsiZW1haWwiOlsicnNqdWxpYW9AZ21haWwuY29tIl19LCJzaWduX2luX3Byb3ZpZGVyIjoicGFzc3dvcmQifX0.ZkBqE8GCSGt9FX_LxoaLNgHcPx19EDMq3ARmZaJ_R1_FiBcQAp8T_AEmleVu68lqw7SdcM2aAjZ1kZbfkZ48hgfhW0LI03VC_6Dc4sq9pgCHWarteCeUz4fE1B6nl4nIbKI3nPQorKYTu82SXEzaRiEwHQCVayiMmnkjzj4d-2YVp4WA8If_h3jNHBe8giskjwkB2t6hB39vYLqvcM5sEeSBRpVT8zA-hmp2AeImcXagCK4Av7JIt_iBNuwT9dwMLtA6addoXcDYTJuRZ3GhVrbL8x_is9u2XDDLWDWdrj1yAjkq7pTPwC7KPft8Md2PKxqYR5bid_VRSjPIeb_k8A

And this Public Key:

Using the site https://jwt.io/, is possible run the signature verify and all it's OK!

Now for better undertand of what is going on I want to gerenate the signature by my self.

My first step was to generate the header using as input this header json:

{"alg":"RS256","kid":"5d340ddbc3c5bacc4cee1fb9d16e9837ec6163eb","typ":"JWT"}

Just translating this with base64UrlEncode and we get:

eyJhbGciOiJSUzI1NiIsImtpZCI6IjVkMzQwZGRiYzNjNWJhY2M0Y2VlMWZiOWQxNmU5ODM3ZWM2MTYzZWIiLCJ0eXAiOiJKV1QifQ

Doing the same transformation on json that represents the payload, and we get the correct result!

{"name":"zagalo","iss":"https://securetoken.google.com/profept-3c479","aud":"profept-3c479","auth_time":1666291403,"user_id":"gRmvqXoKrHO9ODKQDBa0V6tZ50K2","sub":"gRmvqXoKrHO9ODKQDBa0V6tZ50K2","iat":1666291403,"exp":1666295003,"email":"rsjuliao@gmail.com","email_verified":false,"firebase":{"identities":{"email":["rsjuliao@gmail.com"]},"sign_in_provider":"password"}}

eyJuYW1lIjoiemFnYWxvIiwiaXNzIjoiaHR0cHM6Ly9zZWN1cmV0b2tlbi5nb29nbGUuY29tL3Byb2ZlcHQtM2M0NzkiLCJhdWQiOiJwcm9mZXB0LTNjNDc5IiwiYXV0aF90aW1lIjoxNjY2MjkxNDAzLCJ1c2VyX2lkIjoiZ1JtdnFYb0tySE85T0RLUURCYTBWNnRaNTBLMiIsInN1YiI6ImdSbXZxWG9LckhPOU9ES1FEQmEwVjZ0WjUwSzIiLCJpYXQiOjE2NjYyOTE0MDMsImV4cCI6MTY2NjI5NTAwMywiZW1haWwiOiJyc2p1bGlhb0BnbWFpbC5jb20iLCJlbWFpbF92ZXJpZmllZCI6ZmFsc2UsImZpcmViYXNlIjp7ImlkZW50aXRpZXMiOnsiZW1haWwiOlsicnNqdWxpYW9AZ21haWwuY29tIl19LCJzaWduX2luX3Byb3ZpZGVyIjoicGFzc3dvcmQifX0

the last part:

According with jwt.io the signature is:

RSASHA256(
  base64UrlEncode(header) + "." +
  base64UrlEncode(payload))

Lets assume that:
H=base64UrlEncode(header)
and
P=base64UrlEncode(payload)
So, SIGNATURE=RSASHA256( H+'.'+P )

How to get the signature? I have been trying all kind of strategy to generate the correct result, but I'm not having success.

What I have to do to get re correct result? My main strategy is to get the result of SHA256 and put this as argument to RSA with my public key above, but the result is not the same. What I'm doing wrong? What I have to learn to solve my lack of knologe?

Solution

It is not possible to generate the signature of a message, when you have only the public key of RSA. You need the private key to generate the signature. The algorithm for the signature using the "RS256" algorithm for JWT is "RSASSA-PKCS1-v1_5 using SHA-256", as defined in RFC 7518 - 3.3. Digital Signature with RSASSA-PKCS1-v1_5:

      +-------------------+---------------------------------+
      | "alg" Param Value | Digital Signature Algorithm     |
      +-------------------+---------------------------------+
      | RS256             | RSASSA-PKCS1-v1_5 using SHA-256 |
      | RS384             | RSASSA-PKCS1-v1_5 using SHA-384 |
      | RS512             | RSASSA-PKCS1-v1_5 using SHA-512 |
      +-------------------+---------------------------------+

The digital signature algorithm "RSASSA-PKCS1-v1_5" itself is defined in RFC 3447 - 8.2. RSASSA-PKCS1-v1_5. The actual signature algorithm is defined in RFC 3447 - 8.2.1 Signature generation operation:

RSASSA-PKCS1-V1_5-SIGN (K, M)

Input:
K        signer's RSA private key
M        message to be signed, an octet string

Output:
S        signature, an octet string of length k, where k is the
         length in octets of the RSA modulus n

As you see, you need the "signer's RSA private key" K to generate the signature for the message M (which would be your JWT header and payload).

You can use the public key only to verify that a given signature is valid, not create new signatures. Only the owner of the private key can do that.