I am trying to do a signature in rust(v1.77.1) with ring, followed by the official document https://docs.rs/ring/latest/ring/signature/index.html like this, this is the main.rs:

use ring::{rand, rsa, signature};

fn main() {
    let result = sign_and_verify_rsa();
    match result {
        Ok(data) => {
            print!("{:?}", data);
        }
        Err(e) => {
            print!("{:?}", e)
        }
    }
}

fn sign_and_verify_rsa() -> Result<(), MyError> {
    // Create an RSA keypair from the DER-encoded bytes. This example uses
    // a 2048-bit key, but larger keys are also supported.
    let private_key_der = "-----BEGIN PRIVATE KEY-----
    MIIBVAIBADANBgkqhkiG9w0BAQEFAASCAT4wggE6AgEAAkEAv2YlvorgVCb5FMzf
    rzP7xkYGh2FIm4J34GIsTikZ9rWMtSJG3a2YD/s5QgFveYsxv1GnixkxqGsKYVsj
    RJQl/QIDAQABAkBGxlTqspvlQkwUvvVv6f8OAFLluLmKl6IQXP0ZIeWx7iiLDM6d
    WBE8hblDr12KYpWuNLn3b1X+HCDvMfHbbiiBAiEA4rVYZhozGKsr93p0xcyflwj+
    /Bnx7+16YeiNgs76vs0CIQDYIObpFg7eaxhsVhFfLt+PohQhhWEsP+SlkD7LezKj
    8QIhALADX8iV2snyS8Zueq7/eYBH6cBNXHuNAxF0/jlq/V6ZAiBeERPvZvZZv+sF
    XAB7mP8cmMKCRHcitzxhAwMRtTE5QQIgCcnXlJ+Odfzy/vJr3BPlDWXJAZBHok54
    UwWW2C74w/Y=
    -----END PRIVATE KEY-----".as_bytes();
    let key_pair = rsa::KeyPair::from_pkcs8(&private_key_der).map_err(|_| MyError::BadPrivateKey)?;

    // Sign the message "hello, world", using PKCS#1 v1.5 padding and the
    // SHA256 digest algorithm.
    const MESSAGE: &'static [u8] = b"hello, world";
    let rng = rand::SystemRandom::new();
    let mut signature = vec![0; key_pair.public().modulus_len()];
    key_pair
        .sign(&signature::RSA_PKCS1_SHA256, &rng, MESSAGE, &mut signature)
        .map_err(|_| MyError::OOM)?;
    let pub_key = "-----BEGIN PUBLIC KEY-----
    MFwwDQYJKoZIhvcNAQEBBQADSwAwSAJBAL9mJb6K4FQm+RTM368z+8ZGBodhSJuC
    d+BiLE4pGfa1jLUiRt2tmA/7OUIBb3mLMb9Rp4sZMahrCmFbI0SUJf0CAwEAAQ==
    -----END PUBLIC KEY-----".as_bytes();
    // Verify the signature.
    let public_key =
        signature::UnparsedPublicKey::new(&signature::RSA_PKCS1_2048_8192_SHA256, pub_key);
    public_key
        .verify(MESSAGE, &signature)
        .map_err(|_| MyError::BadSignature)
}

#[derive(Debug)]
enum MyError {
    BadPrivateKey,
    OOM,
    BadSignature,
}

this code shows:

BadPrivateKey

the private key format is pkcs8, am I missing something? what should I do to make the signature process works? This is the Cargo.toml:

[dependencies]
rsa = "0.5"
ring = "0.17.8"