In .Net, you can create your key pair like this:
public static Tuple<string, string> CreateKeyPair()
{
    CspParameters cspParams = new CspParameters { ProviderType = 1 };

    RSACryptoServiceProvider rsaProvider = new RSACryptoServiceProvider(1024, cspParams);

    string publicKey = Convert.ToBase64String(rsaProvider.ExportCspBlob(false));
    string privateKey = Convert.ToBase64String(rsaProvider.ExportCspBlob(true));

    return new Tuple<string, string>(privateKey, publicKey);
}

You can then use your public key to encrypt a message like so:
public static byte[] Encrypt(string publicKey, string data)
{
    CspParameters cspParams = new CspParameters { ProviderType = 1 };
    RSACryptoServiceProvider rsaProvider = new RSACryptoServiceProvider(cspParams);

    rsaProvider.ImportCspBlob(Convert.FromBase64String(publicKey));

    byte[] plainBytes = Encoding.UTF8.GetBytes(data);
    byte[] encryptedBytes = rsaProvider.Encrypt(plainBytes, false);

    return encryptedBytes;
}

And use your private key to decrypt like this:

public static string Decrypt(string privateKey, byte[] encryptedBytes)
{
    CspParameters cspParams = new CspParameters { ProviderType = 1 };
    RSACryptoServiceProvider rsaProvider = new RSACryptoServiceProvider(cspParams);

    rsaProvider.ImportCspBlob(Convert.FromBase64String(privateKey));

    byte[] plainBytes = rsaProvider.Decrypt(encryptedBytes, false);

    string plainText = Encoding.UTF8.GetString(plainBytes, 0, plainBytes.Length);

    return plainText;
}