Aspects

vs

Exponent

Exponentiation is a mathematical operation, written as b raised to the power of n, involving two numbers, the base b and the exponent (or power) n.

Modulo

The modulo (sometimes called modulus) operation finds the remainder of division of one number by another

Rsa shorter operations

Quality Example
Shorter

"The rsa private exponent may actually be shorter than the modulo"

from question "Does RSA2048 key modulus have to be exactly 2048 bits long?"

Always less

"When i generate rsa key pairs by openssl it seems like private key private exponent is always less than public key modulo"

from question "Is RSA private key less than public?"

Longer

"The private exponent must not be longer than the modulo"

from question "Microsoft RSA CSP key size"

Better

"So no choice of the public exponent for this modulo is better than 19 using the public exponent to decrypt will work for at least half of the messages when eâ² 9 16 and in many cases for almost all the messages when eâ² 1 16"

from question "RSA public exponent is also decrypting message"

Smaller

"I must also add that designing the rsa key so that the private exponent is substantially shorter than the modulo to speed up operations is a security risk if the exponent is smaller than 29 of the modulo length then the key can be cracked"

from question "A factor that impacts RSA encryption/decryption time"

"The private exponent is always smaller than the modulo so you should be able to encrypt it using the raw rsa operation if you make sure to remove the prepended zero"

from question "How to encrypt a RSAKey using another RSAKey?"

"Edit as gregs points out in the comments you cannot be sure that the private exponent of the key you want to encrypt is smaller than the modulo of the key you want to use to encrypt with"

from question "How to encrypt a RSAKey using another RSAKey?"

Bit large number

Quality Example
Lower

"1024 bit private exponent large number lower than the modulo"

from question "Checking RSA key pairs error"

Smaller

"In rsa signing a message m means exponentiation with the private exponent d the result r is the smallest integer 0 and smaller than the modulo n so that"

from question "RSA signature size?"

Others

Quality Example
Value faster

"Using the pow function and passing a modulo value is faster than computing the full exponent and then taking the modulo because the modulo can be applied to the partial products at each stage of the calculation which stops the value from getting too large 10 6 to the power of 10 6 has 6 million decimal digits with a modulo applied at each step the values never have to grow larger than the size of the modulo - about 13 digits in this example"

from question "Finding remainder mod involving exponent and division involving huge numbers"

Normally shorter

"Little wonder you get errors the exponent is normally shorter than the modulo which is always the same size as the key size"