$ openssl genrsa -out rsa_1024_priv.pem

$ openssl pkcs8 -topk8 -inform PEM -in rsa_1024_priv.pem -outform PEM -nocrypt -out rsa_1024_priv_pkcs8.pem

$ openssl rsa -in rsa_1024_priv_pkcs8.pem -pubout -out rsa_1024_pub.peml

public static class RSAHelper
{
    /// <summary>
    /// RSA签名
    /// </summary>
    /// <param name="content">数据</param>
    /// <param name="privateKey">RSA密钥</param>
    /// <returns></returns>
    public static string RsaSign(string content, string privateKey)
    {
        var signer = SignerUtilities.GetSigner("SHA1withRSA");
        //将java格式的rsa密钥转换成.net格式
        var privateKeyParam = (RsaPrivateCrtKeyParameters)PrivateKeyFactory.CreateKey(StrToToHexByte(privateKey));
        signer.Init(true, privateKeyParam);
        var plainBytes = Encoding.UTF8.GetBytes(content);
        signer.BlockUpdate(plainBytes, 0, plainBytes.Length);
        var signBytes = signer.GenerateSignature();
        return ByteToHexStr(signBytes);
    }

    /// <summary>
    /// RSA验签
    /// </summary>
    /// <param name="content">内容</param>
    /// <param name="publicKey">RSA公钥</param>
    /// <param name="signData">签名字段</param>
    /// <returns></returns>
    public static bool VerifySign(string content, string publicKey, string signData)
    {
        try
        {
            var signer = SignerUtilities.GetSigner("SHA1withRSA");
            var publicKeyParam = (RsaKeyParameters)PublicKeyFactory.CreateKey(StrToToHexByte(publicKey));
            signer.Init(false, publicKeyParam);
            var signBytes = StrToToHexByte(signData);
            var plainBytes = Encoding.UTF8.GetBytes(content);
            signer.BlockUpdate(plainBytes, 0, plainBytes.Length);
            var ret = signer.VerifySignature(signBytes);
            return ret;
        }
        catch (Exception ex)
        {
            return false;
        }
    }

    /// <summary>
    /// 字符串转16进制字节数组
    /// </summary>
    /// <param name="hexString"></param>
    /// <returns></returns>
    private static byte[] StrToToHexByte(string hexString)
    {
        hexString = hexString.Replace(" ", "");
        if ((hexString.Length % 2) != 0)
            hexString += " ";
        byte[] returnBytes = new byte[hexString.Length / 2];
        for (int i = 0; i < returnBytes.Length; i++)
            returnBytes[i] = Convert.ToByte(hexString.Substring(i * 2, 2), 16);
        return returnBytes;
    }
    
    /// <summary>
    /// 字节数组转16进制字符串
    /// </summary>
    /// <param name="bytes"></param>
    /// <returns></returns>
    public static string ByteToHexStr(byte[] bytes)
    {
        string returnStr = "";
        if (bytes != null)
        {
            for (int i = 0; i < bytes.Length; i++)
            {
                returnStr += bytes[i].ToString("X2");
            }
        }
        return returnStr;
    }
}

var content = "123";
var privateKey = RSAHelper.ByteToHexStr(Convert.FromBase64String("MIIBVQIBADANBgkqhkiG9w0BAQEFAASCAT8wggE7AgEAAkEA051JxQSrpN2cgI/fbCFjsALy7G055ichin5FF9qZ6VcdOa4/+V80FMLhR6ifRD2Sb/4qR0pMLnfkJadKBFM/QwIDAQABAkBMV3MUk6HEoXpjWwQUQ1tuVTIEH0eDA1zzVKhieaeK6Q1q4CiqJJ3fMkSTxgQZc6Wy11USJa6cRkYul4hsssddccBAiEA9Iiu7kxwbUE3DNnPzYi7st++fyo2ch9Wh2jF9BQB0YMCIQDdiXK/Y7673ucqBZdVpECJgp3DKCKlJPtfpuRmbSIvQQIhAM0IBdSclu+kbKoDvu7QpMCYRbuOA1Sw3fZvbPr4A4ZNAiBxcakpCNLrMcH+as6MNIg34oMXJL5ZAw8WdEgRi2EuAQIhALx6SB/hoTg91dGPd/Ql6pvRQaEG+HWda2yrW8fd41ot"));
var publicKey = RSAHelper.ByteToHexStr(Convert.FromBase64String("MFwwDQYJKoZIhvcNAQEBBQADSwAweniiO2ndANOdScUEq6TdnICP32whY7AC8uxtOeYnIYp+RRfamelXHTmuP/lfNBTC4Ueon0Q9km/+KkdKTC535CWnSgRTP0MCAwEAAQ=="));

var signData = RSAHelper.RsaSign(content, privateKey);
var result= RSAHelper.VerifySign(content, publicKey, signData);