Private key length > public key?

Question

I'm working on rsa private and public key / encryption / decryption / etc using openssl module. But a question is still unanswered : why private key are often (are always, i don't know) longer than public key ?

Is there several answer to this ?

This is my public and private key generated.

Answer 1

why private key are often (are always, i don't know) longer than public key ?

The answer is detailed in PKCS 1 (and friends like RFC 2437 ).

The public key is the pair {e, n} , where e is the public exponent and n is the modulus.

One of the private key representations is the triplet {e, d, n} , where e is the public exponent, d is the private exponent and n is the modulus.

The other private key representations the n-tuple {e, d, n, p, q, dp, dq, qi} , where e is the public exponent; d is the private exponent; n is the modulus; and p and q are the factors of n .

And the remaining are for the Chinese Remainder theorem, which allows a speedup in signatures (I believe). dp is p's exponent, a positive integer such that e(dP) ≅ 1 (mod(p-1)) ; dq is q's exponent, a positive integer such that e(dq) ≅ 1 (mod(q-1)) ; and qi is CRT coefficient, a positive integer less than p such that q(qInv) ≅ 1 (mod p) .

Answer 2

The RSA private key includes the values that are in the RSA public key, and some more values that are private . Futhermore, the values themselves are larger.

The public key contains the modulus and the public exponent e which is short (mostly 3 or 65537). On the other hand the private key contains the private exponent d ≡ e ⁻¹ (mod φ( n )). Depending on φ( n ), d can be vastly larger than e .

There are other public key algorithms where this is different. Some algorithms based on elliptic curves have a single big integer as the private key (and group description) and the public key has a curve point (and group description) which is larger than the big integer.