RSA private exponent is much larger than RSA public exponent. For example, for a 2048 bit RSA private key, the private exponent can have more than 2000 bits. But the public exponent is usually 65537 (0x10001) which has a much shorter bit length.
Here is my guess of the reason. Let's use define the following symbols to describe RSA algorithm:
- $n$: modulus
- $e$: public exponent
- $d$: private exponent
- $P$: public key $(e,n)$
- $S$: private key $(d,n)$
Encrypt a plain message $M$ with $M^e \text{ mod }n$. Decrypt a encrypted message $C$ with $C^d \text{ mod }n$. Since $d$ is much larger than $e$, decryption incurs much more mulipliation compuatuation. So the use of a large priavate exponent is to make the decryption harder.
Is my guess correct?