# The RSA Algorithm

Soon after Merkle's knapsack algorithm came the first full-fledged public-key algorithm, one that works for encryption and digital signatures: RSA [1328,1329]. Of all the public-key algorithms proposed over the years, RSA is by far the easiest to understand and implement. (Martin Gardner published an early description of the algorithm in his "Mathematical Games" column in Scientific American [599].) It is also the most popular. Named after the three inventors -- Ron Rivest, Adi Shamir, and Leonard Adleman -- it has since withstood years of extensive cryptanalysis. Although the cryptanalysis neither proved nor disproved RSA's security, it does suggest a confidence level in the algorithm.

RSA gets its security from the difficulty of factoring large numbers. The public and private keys are functions of a pair of large (100 to 200 digits or even larger) prime numbers. Recovering the plaintext from the public key and the ciphertext is conjectured to be equivalent to factoring the product of the two primes.

To generate the two keys, choose two random large prime numbers, p and q. For maximum security, choose p and q of equal length. Compute the product:

           n = pq

Then randomly choose the encryption key, e, such that e and (p - 1)(q - 1) are relatively prime. Finally, use the extended Euclidean algorithm to compute the decryption key, d, such that

           ed =- 1 (mod (p - 1)(q - 1))

[Scanner's note: the above equals was an equivalence] In other words,

           d = e-1 mod ((p - 1)(q - 1))

Note that d and n are also relatively prime. The numbers e and n are the public key; the number d is the private key. The two primes, p and q, are no longer needed. They should be discarded, but never revealed.

To encrypt a message m, first divide it into numerical blocks smaller than n (with binary data, choose the largest power of 2 less than n). That is, if both p and q are 100-digit primes, then n will have just under 200 digits and each message block, m, should be just under 200 digits long. (If you need to encrypt a fixed number of blocks, you can pad them with a few zeros on the left to ensure that they will always be less than n. The encrypted message, c, will be made up of similarly sized message blocks, ci, of about the same length. The encryption formula is simply

             ci = mie mod n

To decrypt a message, take each encrypted block ci and compute

             mi = cid mod n

Since

    	cid	= ( mie ) d

    		= mied

    		= mik(p - 1)(q - 1) + 1

    		= mimik(p - 1)(q - 1)

    		= mi * 1

    		= mi ; all ( mod n )

the formula recovers the message. This is summarized in Table 19.2.

Table 19.2 RSA Encryption

Public Key:
n product of two primes, p and q (p and q must remain secret)
e relatively prime to (p - 1)(q - 1)
Private Key:
d e-1 mod ((p - l)(q - l))
Encrypting:
c = me mod n
Decrypting:
m = cd mod n

The message could just as easily have been encrypted with d and decrypted with e; the choice is arbitrary. I will spare you the number theory that proves why this works; most current texts on cryptography cover it in detail. A short example will probably go a long way to making this clearer. If p = 47 and q = 71, then

    	n = pq = 3337

The encryption key, e, must have no factors in common with

    	(p - 1)(q - 1) = 46 * 70 = 3220

Choose e (at random) to be 79. In that case

    	d = 79-1 mod 3220 = 1019

This number was calculated using the extended Euclidean algorithm (see Section 11.3). Publish e and n, and keep d secret. Discard p and q.

To encrypt the message

    	m = 6882326879666683

first break it into small blocks. Three-digit blocks work nicely in this case. The message is split into six blocks, mi, in which

    	m1 = 688
    	m2 = 232
    	m3 = 687
    	m4 = 966
    	m5 = 668
    	m6 = 003

The first block is encrypted as

    	68879 mod 3337 = 1570 = c1

Performing the same operation on the subsequent blocks generates an encrypted message:

    	c = 1570 2756 2091 2276 2423 158

Decrypting the message requires performing the same exponentiation using the decryption key of 1019, so

    	15701019 mod 3337 = 688 = m1

The rest of the message can be recovered in this manner.

This page from Bruce Schneier, Applied Cryptography 2nd Ed, 1996, John Wiley & Sons. Section 19.3, Pages 466 - 469. Used with permission. To order, click here.