Nice resource for ASN.1 structured RSA keys https://etherhack.co.uk/asymmetric/docs/rsa_key_breakdown.html in private key section shows following RSA private key related fields:

  • modulus
  • public exp
  • private expo
  • prime 1
  • prime 2
  • exponent 1
  • exponent 2
  • coefficient

I understand the majority of these fields as per https://en.wikipedia.org/wiki/RSA_(cryptosystem)#Key_generation.

But what are the bolded exponent 1, exponent 2 and coefficient? I cannot find their use in RSA in the linked wiki article.

  • $\begingroup$ Although RSA Labs seems to have disappeared with the acquisition of EMC by Dell, the PKCS1 standard remains available as several RFCs; 8017 sec 3.2 shows the generation process and A.1.2 shows the ASN.1 (used by OpenSSL and many others except 'multiprime' is very rare). $\endgroup$ Nov 16, 2017 at 22:14

1 Answer 1


The are the $d_P$, $d_Q$ and of course $q_{inv}$ - in that order - in the Wikipedia article. As you can see in the formulas relating to the Chinese Remainder Theorem the primes are also used but $d$, the private exponent, isn't.

The CRT calculations can be used to speed up RSA private key operations, which are the slowest part of the RSA cryptosystem after the prime generation. The calculations also have different side channel attacks linked to them compared to "plain" RSA.

  • $\begingroup$ So dp, dq and qinv are used only for ciphertext decryption (using CRT)? $\endgroup$
    – croraf
    Nov 16, 2017 at 20:32
  • 1
    $\begingroup$ Yep, or signature generation, of course, or RSA-KEM... Any private key operation (using CRT). $\endgroup$
    – Maarten Bodewes
    Nov 16, 2017 at 20:48

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