Digital Signing algorithm for a fixed set of private and public keys in broadcasting like system? I have a broadcast kind of system. Where sender knows what to send and receivers will know what sender has sent. Now the sender will sign the message with his own private key and receivers will verify with the provided public key.

Now, all the receivers will have the same public key and the sender will have one protected private key that is being used for signing of the messages.

Keys are not going to be changed for a very long long time. So the senders and receivers will end up having same keys for eternity.

Which cryptographic approach would be suitable for this use case to make sure no one else can sign the message and send to the receivers?
There is a lot of confusion in the question:


message encryption is performed to achieve confidentiality, it is performed using the public key of the receiver's key pair (the public key needs to be trusted by the sender);
message signing is performed to achieve authentication and integrity of messages, it is performed with the private key of the sender's key pair (the public key needs to be trusted by the receiver);


Now you are missing the receiver's key pair in your question. This makes it impossible to encrypt messages for the receiver, as you cannot establish the identity of the entity that is trying to contact you. You could still use Diffie-Hellman key agreement to establish a secret key to prevent passive attacks, but the scheme would not be secure against active attacks.

Public keys are usually not protected against attack. For instance, an RSA public key exponent is usually set to 65537 so anybody that also processes the modulus can guess the public key using only a single guess. Public key operations are usually not protected against leaking the public key either. So any scheme that requires the public key to be secret is build on quicksand.

You need to provide the receiver with a shared secret key or a key pair with a trusted public key to create a secure scheme.



[UPDATE] After the update only message authentication and integrity seem to be requested. This can be performed using any secure digital signature format. RSA with a large key size and PKCS#1 v2.1 PSS padding would likely be the best choice.

Although key rotation is always preferable, a large enough key pair is likely to be secure enough unless quantum computers start to scale really well. Currently QC secure algorithms are not standardized all to well, so it's not like you have much of a choice in that regard. ECDSA could also be used.

It's probably best to use an existing container format for this. Usually CMS would be recommended as protocol.