Eu queria perguntar como recuperar os valores reais das chaves privadas e públicas do objeto KeyPair porque eu preciso exportá-los e salvar em um banco de dados.

KeyPairGenerator kpg = KeyPairGenerator.getInstance("RSA");
kpg.initialize(1024);
KeyPair keyPair = kpg.genKeyPair();

System.out.println("Public key " + keyPair.getPublic());
System.out.println("Private key " + keyPair.getPrivate());

A saída é:

Public key Sun RSA public key, 1024 bits
  modulus: 105712092415375085805423498639048173422142354311030811647243014925610093650322108853068042919471115278002432342007597147610508132502035047888382465733153739247741208519707861808073276783311634229563965825609200080862631487160732889423591650215084096832366499080850540875321197564283324922935557797293830551071
  public exponent: 65537
Private key sun.security.rsa.RSAPrivateCrtKeyImpl@35e71