# RSA

In cryptography, RSA is an algorithm for public-key encryption. It was the first algorithm known to be suitable for signing as well as encryption, and one of the first great advances in public key cryptography. RSA is still widely used in electronic commerce protocols, and is believed to be secure given sufficiently long keys.

## History of RSA

The algorithm was described in 1977 by Ron Rivest, Adi Shamir and Len Adleman at MIT; the letters RSA are the initials of their surnames.

Clifford Cocks, a British mathematician working for GCHQ, described an equivalent system in an internal document in 1973. Given the relatively expensive computers needed to implement it at the time it was mostly considered a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its top-secret classification.

The algorithm was patented by MIT in 1983 in the United States of America as Template:US patent. It expired on 21 September 2000. Since the algorithm had been published prior to patent application, regulations in much of the rest of the world precluded patents elsewhere. Had Cocks' work been publicly known, a patent in the US would not have been possible either.

## Operation

### Key generation

Suppose a user Alice wishes to allow Bob to send her a private message over an insecure transmission medium. She takes the following steps to generate a public key and a private key:

1. Choose two large prime numbers $p\,$ and $q\,$ such that $p\neq q$ , randomly and independently of each other.
2. Compute $n=pq\,$ .
3. Compute the totient $\phi (n)=(p-1)(q-1)\,$ .
4. Choose an integer $e$ such that $1 which is coprime to $\phi (n)\,$ .
5. Compute $d$ such that $de\equiv 1{\pmod {\phi (n)}}$ .
• prime numbers can be probabilistically tested for using Fermat's little theorem: $a^{(p-1)}\equiv 1{\pmod {p}}$ , if $p\,$ is prime; testing with a few values for $a\,$ gives an excellent probability that $p\,$ is prime. (Carmichael numbers are composite numbers that can pass the test for all $a$ with $\gcd(a,n)=1$ , but they are exceedingly rare.)
• (Steps 4 and 5 can be performed with the extended Euclidean algorithm; see modular arithmetic.)
• (Step 5, rewritten, can also be found by finding integer $x\,$ which causes $d={\frac {x(p-1)(q-1)+1}{e}}$ to be an integer, then using the value of $d{\pmod {(p-1)(q-1)}}\,$ ;
• (From step 2 PKCS#1 v2.1 uses $\lambda =LCM(p-1,q-1)\,$ instead of $\phi =(p-1)(q-1)\,$ ).

The public key consists of

• n, the modulus, and
• e, the public exponent (sometimes encryption exponent).

The private key consists of

• n, the modulus, which is public and appears in the public key, and
• d, the private exponent (sometimes decryption exponent), which must be kept secret.

Usually, a different form of the private key (including CRT parameters) is stored:

• p and q, the primes from the key generation,
• d mod (p-1) and d mod (q-1) (often known as dmp1 and dmq1)
• (1/q) mod p (often known as iqmp)

This form allows faster decryption and signing using the Chinese Remainder Theorem (CRT). In this form, all of the parts of the private key must be kept secret.

Alice transmits the public key to Bob, and keeps the private key secret. p and q are sensitive since they are the factors of n, and allow computation of d given e. If p and q are not stored in the CRT form of the private key, they are securely deleted along with the other intermediate values from the key generation.

### Encrypting messages

Suppose Bob wishes to send a message M to Alice. He turns M into a number m < n, using some previously agreed-upon reversible protocol known as a padding scheme.

Bob now has m, and knows n and e, which Alice has announced. He then computes the ciphertext c corresponding to m:

$c=m^{e}\mod {n}$ This can be done quickly using the method of exponentiation by squaring. Bob then transmits c to Alice.

### Decrypting messages

Alice receives c from Bob, and knows her private key d. She can recover m from c by the following procedure:

$m=c^{d}\mod {n}$ Given m, she can recover the original message M. The decryption procedure works because

$c^{d}\equiv (m^{e})^{d}\equiv m^{ed}{\pmod {n}}$ .

Now, since ed ≡ 1 (mod p-1) and ed ≡ 1 (mod q-1), Fermat's little theorem yields

$m^{ed}\equiv m{\pmod {p}}$ and

$m^{ed}\equiv m{\pmod {q}}$ Since p and q are distinct prime numbers, applying the Chinese remainder theorem to these two congruences yields

$m^{ed}\equiv m{\pmod {pq}}$ .

Thus,

$c^{d}\equiv m{\pmod {n}}$ .

### A working example

Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but you can also use OpenSSL to generate and examine a keypair.

We let

 p = 61 — first prime number (to be kept secret or deleted securely) q = 53 — second prime number (to be kept secret or deleted securely) n = pq = 3233 — modulus (to be made public) e = 17 — public exponent (to be made public) d = 2753 — private exponent (to be kept secret)

The public key is (e, n). The private key is d. The encryption function is:

encrypt(m) = me mod n = m17 mod 3233

where m is the plaintext. The decryption function is:

decrypt(c) = cd mod n = c2753 mod 3233

where c is the ciphertext.

To encrypt the plaintext value 123, we calculate

encrypt(123) = 12317 mod 3233 = 855

To decrypt the ciphertext value 855, we calculate

decrypt(855) = 8552753 mod 3233 = 123

Both of these computations can be done efficiently using the square-and-multiply algorithm for modular exponentiation.

When used in practice, RSA must be combined with some form of padding scheme, so that no values of M result in insecure ciphertexts. RSA used without padding may suffer from a number of potential problems:

• The values m = 0 or m = 1 always produce ciphertexts equal to 0 or 1 respectively, due to the properties of exponentiation.
• When encrypting with low encryption exponents (e.g., e = 3) and small values of the m, the (non-modular) result of $m^{e}$ may be strictly less than the modulus n. In this case, ciphertexts may be easily decrypted by taking the eth root of the ciphertext with no regard to the modulus.
• Because RSA encryption is a deterministic encryption algorithm-- i.e., has no random component-- an attacker can successfully launch a chosen plaintext attack against the cryptosystem, building a dictionary by encrypting likely plaintexts under the public key, and storing the resulting ciphertexts. When matching ciphertexts are observed on a communication channel, the attacker can use this dictionary in order to learn the content of the message.

In practice, the first two problems might arise when sending short ASCII messages, where m is the concatenation of one or more ASCII-encoded character(s). A message consisting of a single ASCII NUL character (whose numeric value is 0) would be encoded as m = 0, which produces a ciphertext of 0 regardless of what e and N are used. Likewise, a single ASCII SOH (whose numeric value is 1) would always produce a ciphertext of 1. For systems which conventionally use small values of e, such as 3, all single character ASCII messages encoded using this scheme would be insecure, since the largest m would have a value of 255, and 2553 is less than any reasonable modulus. Such plaintexts could be recovered by simply taking the cube root of the ciphertext.

To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. This padding ensures that m does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts. The latter property can increase the cost of a dictionary attack beyond the capabilities of a reasonable attacker.

Standards such as PKCS have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext m with some number of additional bits, the size of the un-padded message M must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks which may be facilitated by a predictable message structure. Early versions of the PKCS standard used ad-hoc constructions, which were later found vulnerable to a practical adaptive chosen ciphertext attack. Modern constructions use secure techniques such as Optimal Asymmetric Encryption Padding (OAEP) to protect messages while preventing these attacks. The PKCS standard also incorporates processing schemes designed to provide additional security for RSA signatures, e.g., the Probabilistic Signature Scheme for RSA (RSA-PSS).

### Signing messages

RSA can also be used to sign a message. Suppose Alice wishes to send a signed message to Bob. She produces a hash value of the message, raises it to the power of d mod n (as she does when decrypting a message), and attaches it as a "signature" to the message. When Bob receives the signed message, he raises the signature to the power of e mod n (as he does when encrypting a message), and compares the resulting hash value with the message's actual hash value. If the two agree, he knows that the author of the message was in possession of Alice's secret key, and that the message has not been tampered with since.

Note that secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption, and that the same key should never be used for both encryption and signing purposes.

## Security

The security of the RSA cryptosystem is based on two mathematical problems: the problem of factoring very large numbers, and the RSA problem. Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are hard, i.e., no efficient algorithm exists for solving them. Providing security against partial decryption may require the addition of a secure padding scheme.

The RSA problem is defined as the task of taking eth roots modulo a composite n: recovering a value m such that me=c mod n, where (e, n) is an RSA public key and c is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus n. With the ability to recover prime factors, an attacker can compute the secret exponent d from a public key (e, n), then decrypt c using the standard procedure. To accomplish this, an attacker factors n into p and q, and computes (p-1)(q-1) which allows the determination of d from e. No polynomial-time method for factoring large integers on a classical computer has yet been found, but it has not been proven that none exists. See integer factorization for a discussion of this problem.

As of 2005, the largest number factored by general-purpose methods was 663 bits long, using state-of-the-art distributed methods. RSA keys are typically 1024–2048 bits long. Some experts believe that 1024-bit keys may become breakable in the near term (though this is disputed); few see any way that 4096-bit keys could be broken in the foreseeable future. Therefore, it is generally presumed that RSA is secure if n is sufficiently large. If n is 256 bits or shorter, it can be factored in a few hours on a personal computer, using software already freely available. If n is 512 bits or shorter, it can be factored by several hundred computers as of 1999. A theoretical hardware device named TWIRL and described by Shamir and Tromer in 2003 called into question the security of 1024 bit keys. It is currently recommended that n be at least 2048 bits long.

In 1993, Peter Shor published Shor's algorithm, showing that a quantum computer could in principle perform the factorization in polynomial time, rendering RSA and related algorithms obsolete. However, quantum computation is not expected to be developed to such a level until at least 2015 or beyond.

## Practical considerations

### Key generation

Finding the large primes p and q is usually done by testing random numbers of the right size with probabilistic primality tests which quickly eliminate virtually all non-primes.

p and q should not be 'too close', lest the Fermat factorization for n be successful. Furthermore, if either p-1 or q-1 has only small prime factors, n can be factored quickly and these values of p or q should therefore be discarded as well.

One should not employ a prime search method which gives any information whatsoever about the primes to the attacker. In particular, a good random number generator for the start value needs to be employed. Note that the requirement here is both 'random' and 'unpredictable'. These are not the same criteria; a number may have been chosen by a random process (ie, no pattern in the results), but if it is predictable in any manner (or even partially predicatable), the method used will result in loss of security. For example, the random number table published by the Rand Corp in the 1950s might very well be truly random, but it has been published and thus can serve an attacker as well. If the attacker can guess half of the digits of p or q, they can quickly compute the other half (shown by Coppersmith in 1997).

It is important that the secret key d be large enough. Wiener showed in 1990 that if p is between q and 2q (which is quite typical) and d < n1/4/3, then d can be computed efficiently from n and e. The encryption exponent e = 2 should also not be used.

### Speed

RSA is much slower than DES and other symmetric cryptosystems. In practice, Bob typically encrypts a secret message with a symmetric algorithm, encrypts the (comparatively short) symmetric key with RSA, and transmits both the RSA-encrypted symmetric key and the symmetrically-encrypted message to Alice.

This procedure raises additional security issues. For instance, it is of utmost importance to use a strong random number generator for the symmetric key, because otherwise Eve could bypass RSA by guessing the symmetric key.

### Key distribution

As with all ciphers, how RSA public keys are distributed is important to security. Key distribution must be secured against a man-in-the-middle attack. Suppose Eve has some way to give Bob arbitrary keys and make him believe they belong to Alice. Suppose further that Eve can intercept transmissions between Alice and Bob. Eve sends Bob her own public key, which Bob believes to be Alice's. Eve can then intercept any ciphertext sent by Bob, decrypt it with her own secret key, keep a copy of the message, encrypt the message with Alice's public key, and send the new ciphertext to Alice. In principle, neither Alice nor Bob would be able to detect Eve's presence. Defenses against such attacks are often based on digital certificates or other components of a public key infrastructure.

### Timing attacks

Kocher described an ingenious new attack on RSA in 1995: if the attacker Eve knows Alice's hardware in sufficient detail and is able to measure the decryption times for several known ciphertexts, she can deduce the decryption key d quickly. This attack can also be applied against the RSA signature scheme. In 2003, Boneh and Brumley demonstrated a more practical attack capable of recovering RSA factorizations over a network connection (e.g., from a Secure Socket Layer (SSL)-enabled webserver). This attack takes advantage of information leaked by the Chinese Remainder Theorem optimization used by many RSA implementations.

One way to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext. However, this approach can significantly reduce performance. Instead, most RSA implementations use an alternate technique known as cryptographic blinding. RSA blinding makes use of the multiplicative property of RSA. Instead of computing cd mod n, Alice first chooses a secret random value r and computes (rec)d mod n. The result of this computation is rm mod n and so the effect of r can be removed by multiplying by its inverse. A new value of r is chosen for each ciphertext. With blinding applied, the decryption time is no longer correlated to the value of the input ciphertext and so the timing attack fails.