RC4

For the Vietnam road named RC4, see Battle of Route Coloniale 4.

RC4

General
Designers	Ron Rivest (RSA Security)
First published	Leaked in 1994 (designed in 1987)
Cipher detail
Key sizes	40–2048 bits
State size	2048 bits
Rounds	256
Speed	7 cycles per byte on original Pentium

In cryptography, RC4 (also known as ARC4 or ARCFOUR meaning Alleged RC4, see below) is the most widely-used software stream cipher and is used in popular protocols such as Secure Sockets Layer (SSL) (to protect Internet traffic) and WEP (to secure wireless networks). While remarkable for its simplicity and speed in software, RC4 has weaknesses that argue against its use in new systems. It is especially vulnerable when the beginning of the output keystream is not discarded, or nonrandom or related keys are used; some ways of using RC4 can lead to very insecure cryptosystems such as WEP.

History

RC4 was designed by Ron Rivest of RSA Security in 1987. While it is officially termed "Rivest Cipher 4", the RC acronym is alternatively understood to stand for "Ron's Code" (see also RC2, RC5 and RC6).

RC4 was initially a trade secret, but in September 1994 a description of it was anonymously posted to the Cypherpunks mailing list. It was soon posted on the sci.crypt newsgroup, and from there to many sites on the Internet. The leaked code was confirmed to be genuine as its output was found to match that of proprietary software using licensed RC4. Because the algorithm is known, it is no longer a trade secret. The name "RC4" is trademarked, so RC4 is often referred to as "ARCFOUR" or "ARC4" (meaning Alleged RC4) to avoid trademark problems. RSA Security has never officially released the algorithm; Rivest has, however, linked to the Wikipedia article on RC4 in his own course notes. RC4 has become part of some commonly-used encryption protocols and standards, including WEP and WPA for wireless cards and TLS.

The main factors in RC4's success over such a wide range of applications are its speed and simplicity: efficient implementations in both software and hardware are very easy to develop.

Description

RC4 generates a pseudorandom stream of bits (a keystream). As with any stream cipher, these can be used for encryption by combining it with the plaintext using bit-wise exclusive-or; decryption is performed the same way (since exclusive-or is a symmetric operation). (This is similar to the Vernam cipher except that generated pseudorandom bits, rather than a prepared stream, are used.) To generate the keystream, the cipher makes use of a secret internal state which consists of two parts:

1. A permutation of all 256 possible bytes (denoted "S" below).
2. Two 8-bit index-pointers (denoted "i" and "j").

The permutation is initialized with a variable length key, typically between 40 and 256 bits, using the key-scheduling algorithm (KSA). Once this has been completed, the stream of bits is generated using the pseudo-random generation algorithm (PRGA).

The key-scheduling algorithm (KSA)

The key-scheduling algorithm is used to initialize the permutation in the array "S". "keylength" is defined as the number of bytes in the key and can be in the range 1 ≤ keylength ≤ 256, typically between 5 and 16, corresponding to a key length of 40 – 128 bits. First, the array "S" is initialized to the identity permutation. S is then processed for 256 iterations in a similar way to the main PRGA, but also mixes in bytes of the key at the same time.

for i from 0 to 255
    S[i] := i
endfor
j := 0
for i from 0 to 255
    j := (j + S[i] + key[i mod keylength]) mod 256
    swap values of S[i] and S[j]
endfor

The pseudo-random generation algorithm (PRGA)

The lookup stage of RC4. The output byte is selected by looking up the values of S(i) and S(j), adding them together modulo 256, and then looking up the sum in S; S(S(i) + S(j)) is used as a byte of the key stream, K.

For as many iterations as are needed, the PRGA modifies the state and outputs a byte of the keystream. In each iteration, the PRGA increments i, adds the value of S pointed to by i to j, exchanges the values of S[i] and S[j], and then outputs the element of S at the location S[i] + S[j] (modulo 256). Each element of S is swapped with another element at least once every 256 iterations.

i := 0
j := 0
while GeneratingOutput:
    i := (i + 1) mod 256
    j := (j + S[i]) mod 256
    swap values of S[i] and S[j]
    K := S[(S[i] + S[j]) mod 256]  
    byte_ciphered := K XOR byte_message
endwhile

Implementation

Many stream ciphers are based on linear feedback shift registers (LFSRs), which while efficient in hardware are less so in software. The design of RC4 avoids the use of LFSRs, and is ideal for software implementation, as it requires only byte manipulations. It uses 256 bytes of memory for the state array, S[0] through S[255], k bytes of memory for the key, key[0] through key[k-1], and integer variables, i, j, and y. Performing a modulus 256 can be done with a bitwise AND with 255 (or on most platforms, simple addition of bytes ignoring overflow).

Here is a simple implementation in C:

unsigned char S[256];
unsigned int i, j;

void swap(unsigned char *s, unsigned int i, unsigned int j) {
    unsigned char temp = s[i];
    s[i] = s[j];
    s[j] = temp;
}

/* KSA */
void rc4_init(unsigned char *key, unsigned int key_length) {
    for (i = 0; i < 256; i++)
        S[i] = i;
 
    for (i = j = 0; i < 256; i++) {
        j = (j + key[i % key_length] + S[i]) & 255;
        swap(S, i, j);
    }
 
    i = j = 0;
}
 
/* PRGA */
unsigned char rc4_output() {
    i = (i + 1) & 255;
    j = (j + S[i]) & 255;

    swap(S, i, j);

    return S[(S[i] + S[j]) & 255];
}
 
#include <stdio.h>
#include <string.h>
#include <stdlib.h>

#define ARRAY_SIZE(a) (sizeof(a)/sizeof(a[0]))

int main() {
    unsigned char *test_vectors[][2] = 
    {
        {"Key", "Plaintext"},
        {"Wiki", "pedia"},
        {"Secret", "Attack at dawn"}
    };
 
    int x;
    for (x = 0; x < ARRAY_SIZE(test_vectors); x++) {
        int y;
        rc4_init(test_vectors[x][0], strlen((char*)test_vectors[x][0]));

        for (y = 0; y < strlen((char*)test_vectors[x][1]); y++)
            printf("%02X", test_vectors[x][1][y] ^ rc4_output());
        printf("\n");
    }
    return 0;
}

Test vectors

These test vectors are not official, but convenient for anyone testing their own RC4 program. The keys and plaintext are ASCII, the ciphertext is in hexadecimal.

Key	Keystream	Plaintext	Ciphertext
Key	eb9f7781b734ca72a719...	Plaintext	BBF316E8D940AF0AD3
Wiki	6044db6d41b7...	pedia	1021BF0420
Secret	04d46b053ca87b59...	Attack at dawn	45A01F645FC35B383552544B9BF5

Security

Unlike a modern stream cipher (such as those in eSTREAM), RC4 does not take a separate nonce alongside the key. This means that if a single long-term key is to be used to securely encrypt multiple streams, the cryptosystem must specify how to combine the nonce and the long-term key to generate the stream key for RC4. One approach to addressing this is to generate a "fresh" RC4 key by hashing a long-term key with a nonce. However, many applications that use RC4 simply concatenate key and nonce; RC4's weak key schedule then gives rise to a variety of serious problems.

Like any other stream cipher, RC4 is malleable, so it must be used together with a strong message authentication code. Failure to do so makes it vulnerable to a bit-flipping attack.

Roos' Biases and Key Reconstruction from Permutation

In 1995, Andrew Roos experimentally observed that the first byte of the keystream is correlated to the first three bytes of the key and the first few bytes of the permutation after the KSA are correlated to some linear combination of the key bytes. These biases remained unproved until 2007, when Paul, Rathi and Maitra proved the keystream-key correlation and Paul and Maitra proved the permutation-key correlations. The latter work also used Roos' permutation-key correlations to design the first algorithm for complete key reconstruction from the final permutation after the KSA, without any assumption on the key or IV. This algorithm has a constant probability of success in a time which is the square root of the exhaustive key search complexity. Subsequently, many other works have been done on key reconstruction from RC4 internal states. In another work, Maitra and Paul showed that the Roos type biases still persist even when one considers nested permutation indices, like S[S[i]] or S[S[S[i]]]. These types of biases are used in some of the later key reconstruction methods for increasing the success probability.

Biased Outputs of the RC4

The keystream generated by the RC4 is biased in varying degrees towards certain sequences. The best such attack is due to Itsik Mantin and Adi Shamir who showed that the second output byte of the cipher was biased toward zero with probability 1/128 (instead of 1/256). This is due to the fact that if the third byte of the original state is zero, and the second byte is not equal to 2, then the second output byte is always zero. Such bias can be detected by observing only 256 bytes.

Souradyuti Paul and Bart Preneel of COSIC showed that the first and the second bytes of the RC4 were also biased. The number of required samples to detect this bias is 2²⁵ bytes.

Fluhrer and McGrew also showed such attacks which distinguished the keystream of the RC4 from a random stream given a gigabyte of output.

The complete characterization of a single step of RC4 PRGA was performed by Basu, Ganguly, Maitra and Paul. Considering all the permutations, they prove that the distribution of the output is not uniform given i and j, and as a consequence, information about j is always leaked from the output.

Fluhrer, Mantin and Shamir attack

Main article: Fluhrer, Mantin and Shamir attack

In 2001, a new and surprising discovery was made by Fluhrer, Mantin and Shamir: over all possible RC4 keys, the statistics for the first few bytes of output keystream are strongly non-random, leaking information about the key. If the long-term key and nonce are simply concatenated to generate the RC4 key, this long-term key can be discovered by analysing a large number of messages encrypted with this key. This and related effects were then used to break the WEP ("wired equivalent privacy") encryption used with 802.11 wireless networks. This caused a scramble for a standards-based replacement for WEP in the 802.11 market, and led to the IEEE 802.11i effort and WPA.

Cryptosystems can defend against this attack by discarding the initial portion of the keystream. Such a modified algorithm is traditionally called "RC4-drop[n]", where n is the number of initial keystream bytes that are dropped. The SCAN default is n = 768 bytes, but a conservative value would be n = 3072 bytes.

Klein's Attack

In 2005, Andreas Klein presented an analysis of the RC4 stream cipher showing more correlations between the RC4 keystream and the key. Erik Tews, Ralf-Philipp Weinmann, and Andrei Pychkine used this analysis to create aircrack-ptw, a tool which cracks 104-bit RC4 used in 128-bit WEP in under a minute. Whereas the Fluhrer, Mantin, and Shamir attack used around 10 million messages, aircrack-ptw can break 104-bit keys in 40,000 frames with 50% probability, or in 85,000 frames with 95% probability.

Combinatorial problem

A combinatorial problem related to the number of inputs and outputs of the RC4 cipher was first posed by Itsik Mantin and Adi Shamir in 2001, whereby, of the total 256 elements in the typical state of RC4, if x number of elements (x ≤ 256) are only known (all other elements can be assumed empty), then the maximum number of elements that can be produced deterministically is also x in the next 256 rounds. This conjecture was put to rest in 2004 with a formal proof given by Souradyuti Paul and Bart Preneel.

RC4-based cryptosystems

• WEP
• WPA (default algorithm, but can be configured to use AES-CCMP instead of RC4)
• BitTorrent protocol encryption
• Microsoft Point-to-Point Encryption
• Opera Mini
• Secure Sockets Layer (optionally)
• Secure shell (optionally)
• Remote Desktop Protocol
• Kerberos (optionally)
• SASL Mechanism Digest-MD5 (optionally)
• Gpcode.AK, an early June 2008 computer virus for Microsoft Windows, which takes documents hostage for ransom by obscuring them with RC4 and RSA-1024 encryption
• PDF
• Skype (in modified form)

Where a cryptosystem is marked with "(optionally)", RC4 is one of several ciphers the system can be configured to use.

See also

• eSTREAM - An evaluation of new stream ciphers being conducted by the EU.
• TEA, Block TEA also known as eXtended TEA and Corrected Block TEA - A family of block ciphers that, like RC4, are designed to be very simple to implement.
• Advanced Encryption Standard
• CipherSaber

External links

RC4

• IETF Draft - A Stream Cipher Encryption Algorithm "Arcfour"
• Original posting of RC4 algorithm to Cypherpunks mailing list
• SCAN's entry for RC4
• Attacks on RC4
• RC4 - Cryptology Pointers by Helger Lipmaa
• RSA Security Response to Weaknesses in Key Scheduling Algorithm of RC4
• T-SQL implementation

RC4 in WEP

• (in)Security of the WEP algorithm
• Fluhrer, Mantin, and Shamir attack on WEP (postscript format)