An Overview of Cryptographyby About the author, garykessler.net
January 26th 2012
An Overview of Cryptography
Gary C. Kessler
26 January 2012
Does increased security provide comfort to paranoid people? Or does security provide some very basic protections that we are naive to believe that we don't need? During this time when the Internet provides essential communication between tens of millions of people and is being increasingly used as a tool for commerce, security becomes a tremendously important issue to deal with.
There are many aspects to security and many applications, ranging from secure commerce and payments to private communications and protecting passwords. One essential aspect for secure communications is that of cryptography, which is the focus of this chapter. But it is important to note that while cryptography is necessary for secure communications, it is not by itself sufficient. The reader is advised, then, that the topics covered in this chapter only describe the first of many steps necessary for better security in any number of situations.
This paper has two major purposes. The first is to define some of the terms and concepts behind basic cryptographic methods, and to offer a way to compare the myriad cryptographic schemes in use today. The second is to provide some real examples of cryptography in use today.
I would like to say at the outset that this paper is very focused on terms, concepts, and schemes in current use and is not a treatise of the whole field. No mention is made here about pre-computerized crypto schemes, the difference between a substitution and transposition cipher, cryptanalysis, or other history. Interested readers should check out some of the books in the bibliography below for this detailed — and interesting! — background information.
Cryptography is the science of writing in secret code and is an ancient art; the first documented use of cryptography in writing dates back to circa 1900 B.C. when an Egyptian scribe used non-standard hieroglyphs in an inscription. Some experts argue that cryptography appeared spontaneously sometime after writing was invented, with applications ranging from diplomatic missives to war-time battle plans. It is no surprise, then, that new forms of cryptography came soon after the widespread development of computer communications. In data and telecommunications, cryptography is necessary when communicating over any untrusted medium, which includes just about any network, particularly the Internet.
Within the context of any application-to-application communication, there are some specific security requirements, including:
- Authentication: The process of proving one's identity. (The primary forms of host-to-host authentication on the Internet today are name-based or address-based, both of which are notoriously weak.)
- Privacy/confidentiality: Ensuring that no one can read the message except the intended receiver.
- Integrity: Assuring the receiver that the received message has not been altered in any way from the original.
- Non-repudiation: A mechanism to prove that the sender really sent this message.
Cryptography, then, not only protects data from theft or alteration, but can also be used for user authentication. There are, in general, three types of cryptographic schemes typically used to accomplish these goals: secret key (or symmetric) cryptography, public-key (or asymmetric) cryptography, and hash functions, each of which is described below. In all cases, the initial unencrypted data is referred to as plaintext. It is encrypted into ciphertext, which will in turn (usually) be decrypted into usable plaintext.
In many of the descriptions below, two communicating parties will be referred to as Alice and Bob; this is the common nomenclature in the crypto field and literature to make it easier to identify the communicating parties. If there is a third or fourth party to the communication, they will be referred to as Carol and Dave. Mallory is a malicious party, Eve is an eavesdropper, and Trent is a trusted third party.
There are several ways of classifying cryptographic algorithms. For purposes of this paper, they will be categorized based on the number of keys that are employed for encryption and decryption, and further defined by their application and use. The three types of algorithms that will be discussed are (Figure 1):
- Secret Key Cryptography (SKC): Uses a single key for both encryption and decryption
- Public Key Cryptography (PKC): Uses one key for encryption and another for decryption
- Hash Functions: Uses a mathematical transformation to irreversibly "encrypt" information
With secret key cryptography, a single key is used for both encryption and decryption. As shown in Figure 1A, the sender uses the key (or some set of rules) to encrypt the plaintext and sends the ciphertext to the receiver. The receiver applies the same key (or ruleset) to decrypt the message and recover the plaintext. Because a single key is used for both functions, secret key cryptography is also called symmetric encryption.
With this form of cryptography, it is obvious that the key must be known to both the sender and the receiver; that, in fact, is the secret. The biggest difficulty with this approach, of course, is the distribution of the key.
Secret key cryptography schemes are generally categorized as being either stream ciphers or block ciphers. Stream ciphers operate on a single bit (byte or computer word) at a time and implement some form of feedback mechanism so that the key is constantly changing. A block cipher is so-called because the scheme encrypts one block of data at a time using the same key on each block. In general, the same plaintext block will always encrypt to the same ciphertext when using the same key in a block cipher whereas the same plaintext will encrypt to different ciphertext in a stream cipher.
Stream ciphers come in several flavors but two are worth mentioning here. Self-synchronizing stream ciphers calculate each bit in the keystream as a function of the previous n bits in the keystream. It is termed "self-synchronizing" because the decryption process can stay synchronized with the encryption process merely by knowing how far into the n-bit keystream it is. One problem is error propagation; a garbled bit in transmission will result in n garbled bits at the receiving side. Synchronous stream ciphers generate the keystream in a fashion independent of the message stream but by using the same keystream generation function at sender and receiver. While stream ciphers do not propagate transmission errors, they are, by their nature, periodic so that the keystream will eventually repeat.
Block ciphers can operate in one of several modes; the following four are the most important:
- Electronic Codebook (ECB) mode is the simplest, most obvious application: the secret key is used to encrypt the plaintext block to form a ciphertext block. Two identical plaintext blocks, then, will always generate the same ciphertext block. Although this is the most common mode of block ciphers, it is susceptible to a variety of brute-force attacks.
- Cipher Block Chaining (CBC) mode adds a feedback mechanism to the encryption scheme. In CBC, the plaintext is exclusively-ORed (XORed) with the previous ciphertext block prior to encryption. In this mode, two identical blocks of plaintext never encrypt to the same ciphertext.
- Cipher Feedback (CFB) mode is a block cipher implementation as a self-synchronizing stream cipher. CFB mode allows data to be encrypted in units smaller than the block size, which might be useful in some applications such as encrypting interactive terminal input. If we were using 1-byte CFB mode, for example, each incoming character is placed into a shift register the same size as the block, encrypted, and the block transmitted. At the receiving side, the ciphertext is decrypted and the extra bits in the block (i.e., everything above and beyond the one byte) are discarded.
- Output Feedback (OFB) mode is a block cipher implementation conceptually similar to a synchronous stream cipher. OFB prevents the same plaintext block from generating the same ciphertext block by using an internal feedback mechanism that is independent of both the plaintext and ciphertext bitstreams.
A nice overview of these different modes can be found at progressive-coding.com.
Secret key cryptography algorithms that are in use today include:
Data Encryption Standard (DES): The most common SKC scheme used today, DES was designed by IBM in the 1970s and adopted by the National Bureau of Standards (NBS) [now the National Institute for Standards and Technology (NIST)] in 1977 for commercial and unclassified government applications. DES is a block-cipher employing a 56-bit key that operates on 64-bit blocks. DES has a complex set of rules and transformations that were designed specifically to yield fast hardware implementations and slow software implementations, although this latter point is becoming less significant today since the speed of computer processors is several orders of magnitude faster today than twenty years ago. IBM also proposed a 112-bit key for DES, which was rejected at the time by the government; the use of 112-bit keys was considered in the 1990s, however, conversion was never seriously considered.
DES is defined in American National Standard X3.92 and three Federal Information Processing Standards (FIPS):
Information about vulnerabilities of DES can be obtained from the Electronic Frontier Foundation.
Two important variants that strengthen DES are:
Triple-DES (3DES): A variant of DES that employs up to three 56-bit keys and makes three encryption/decryption passes over the block; 3DES is also described in FIPS 46-3 and is the recommended replacement to DES.
DESX: A variant devised by Ron Rivest. By combining 64 additional key bits to the plaintext prior to encryption, effectively increases the keylength to 120 bits.
More detail about DES, 3DES, and DESX can be found below in Section 5.4.
Advanced Encryption Standard (AES): In 1997, NIST initiated a very public, 4-1/2 year process to develop a new secure cryptosystem for U.S. government applications. The result, the Advanced Encryption Standard, became the official successor to DES in December 2001. AES uses an SKC scheme called Rijndael, a block cipher designed by Belgian cryptographers Joan Daemen and Vincent Rijmen. The algorithm can use a variable block length and key length; the latest specification allowed any combination of keys lengths of 128, 192, or 256 bits and blocks of length 128, 192, or 256 bits. NIST initially selected Rijndael in October 2000 and formal adoption as the AES standard came in December 2001. FIPS PUB 197 describes a 128-bit block cipher employing a 128-, 192-, or 256-bit key. The AES process and Rijndael algorithm are described in more detail below in Section 5.9.
CAST-128/256: CAST-128, described in Request for Comments (RFC) 2144, is a DES-like substitution-permutation crypto algorithm, employing a 128-bit key operating on a 64-bit block. CAST-256 (RFC 2612) is an extension of CAST-128, using a 128-bit block size and a variable length (128, 160, 192, 224, or 256 bit) key. CAST is named for its developers, Carlisle Adams and Stafford Tavares and is available internationally. CAST-256 was one of the Round 1 algorithms in the AES process.
International Data Encryption Algorithm (IDEA): Secret-key cryptosystem written by Xuejia Lai and James Massey, in 1992 and patented by Ascom; a 64-bit SKC block cipher using a 128-bit key. Also available internationally.
Rivest Ciphers (aka Ron's Code): Named for Ron Rivest, a series of SKC algorithms.
RC1: Designed on paper but never implemented.
RC2: A 64-bit block cipher using variable-sized keys designed to replace DES. It's code has not been made public although many companies have licensed RC2 for use in their products. Described in RFC 2268.
RC3: Found to be breakable during development.
RC4: A stream cipher using variable-sized keys; it is widely used in commercial cryptography products, although it can only be exported using keys that are 40 bits or less in length.
RC5: A block-cipher supporting a variety of block sizes, key sizes, and number of encryption passes over the data. Described in RFC 2040.
RC6: An improvement over RC5, RC6 was one of the AES Round 2 algorithms.
Blowfish: A symmetric 64-bit block cipher invented by Bruce Schneier; optimized for 32-bit processors with large data caches, it is significantly faster than DES on a Pentium/PowerPC-class machine. Key lengths can vary from 32 to 448 bits in length. Blowfish, available freely and intended as a substitute for DES or IDEA, is in use in over 80 products.
Twofish: A 128-bit block cipher using 128-, 192-, or 256-bit keys. Designed to be highly secure and highly flexible, well-suited for large microprocessors, 8-bit smart card microprocessors, and dedicated hardware. Designed by a team led by Bruce Schneier and was one of the Round 2 algorithms in the AES process.
Camellia: A secret-key, block-cipher crypto algorithm developed jointly by Nippon Telegraph and Telephone (NTT) Corp. and Mitsubishi Electric Corporation (MEC) in 2000. Camellia has some characteristics in common with AES: a 128-bit block size, support for 128-, 192-, and 256-bit key lengths, and suitability for both software and hardware implementations on common 32-bit processors as well as 8-bit processors (e.g., smart cards, cryptographic hardware, and embedded systems). Also described in RFC 3713. Camellia's application in IPsec is described in RFC 4312 and application in OpenPGP in RFC 5581.
MISTY1: Developed at Mitsubishi Electric Corp., a block cipher using a 128-bit key and 64-bit blocks, and a variable number of rounds. Designed for hardware and software implementations, and is resistant to differential and linear cryptanalysis. Described in RFC 2994.
Secure and Fast Encryption Routine (SAFER): Secret-key crypto scheme designed for implementation in software. Versions have been defined for 40-, 64-, and 128-bit keys.
KASUMI: A block cipher using a 128-bit key that is part of the Third-Generation Partnership Project (3gpp), formerly known as the Universal Mobile Telecommunications System (UMTS). KASUMI is the intended confidentiality and integrity algorithm for both message content and signaling data for emerging mobile communications systems.
SEED: A block cipher using 128-bit blocks and 128-bit keys. Developed by the Korea Information Security Agency (KISA) and adopted as a national standard encryption algorithm in South Korea. Also described in RFC 4269.
ARIA: A 128-bit block cipher employing 128-, 192-, and 256-bit keys. Developed by large group of researchers from academic institutions, research institutes, and federal agencies in South Korea in 2003, and subsequently named a national standard. Described in RFC 5794.
CLEFIA: Described in RFC 6114, CLEFIA is a 128-bit block cipher employing key lengths of 128, 192, and 256 bits (which is compatible with AES). The CLEFIA algorithm was first published in 2007 by Sony Corporation. CLEFIA is one of the new-generation lightweight blockcipher algorithms designed after AES, offering high performance in software and hardware as well as a lightweight implementation in hardware.
SMS4: SMS4 is a 128-bit block cipher using 128-bit keys and 32 rounds to process a block. Declassified in 2006, SMS4 is used in the Chinese National Standard for Wireless Local Area Network (LAN) Authentication and Privacy Infrastructure (WAPI). SMS4 had been a proposed cipher for the Institute of Electrical and Electronics Engineers (IEEE) 802.11i standard on security mechanisms for wireless LANs, but has yet to be accepted by the IEEE or International Organization for Standardization (ISO). SMS4 is described in SMS4 Encryption Algorithm for Wireless Networks (translated and typeset by Whitfield Diffie and George Ledin, 2008) or in the original Chinese.
Skipjack: SKC scheme proposed for Capstone. Although the details of the algorithm were never made public, Skipjack was a block cipher using an 80-bit key and 32 iteration cycles per 64-bit block.
Public-key cryptography has been said to be the most significant new development in cryptography in the last 300-400 years. Modern PKC was first described publicly by Stanford University professor Martin Hellman and graduate student Whitfield Diffie in 1976. Their paper described a two-key crypto system in which two parties could engage in a secure communication over a non-secure communications channel without having to share a secret key.
PKC depends upon the existence of so-called one-way functions, or mathematical functions that are easy to computer whereas their inverse function is relatively difficult to compute. Let me give you two simple examples:
- Multiplication vs. factorization: Suppose I tell you that I have two numbers, 9 and 16, and that I want to calculate the product; it should take almost no time to calculate the product, 144. Suppose instead that I tell you that I have a number, 144, and I need you tell me which pair of integers I multiplied together to obtain that number. You will eventually come up with the solution but whereas calculating the product took milliseconds, factoring will take longer because you first need to find the 8 pairs of integer factors and then determine which one is the correct pair.
- Exponentiation vs. logarithms: Suppose I tell you that I want to take the number 3 to the 6th power; again, it is easy to calculate 36=729. But if I tell you that I have the number 729 and want you to tell me the two integers that I used, x and y so that logx 729 = y, it will take you longer to find all possible solutions and select the pair that I used.
While the examples above are trivial, they do represent two of the functional pairs that are used with PKC; namely, the ease of multiplication and exponentiation versus the relative difficulty of factoring and calculating logarithms, respectively. The mathematical "trick" in PKC is to find a trap door in the one-way function so that the inverse calculation becomes easy given knowledge of some item of information. (The problem is further exacerbated because the algorithms don't use just any old integers, but very large prime numbers.)
Generic PKC employs two keys that are mathematically related although knowledge of one key does not allow someone to easily determine the other key. One key is used to encrypt the plaintext and the other key is used to decrypt the ciphertext. The important point here is that it does not matter which key is applied first, but that both keys are required for the process to work (Figure 1B). Because a pair of keys are required, this approach is also called asymmetric cryptography.
In PKC, one of the keys is designated the public key and may be advertised as widely as the owner wants. The other key is designated the private key and is never revealed to another party. It is straight forward to send messages under this scheme. Suppose Alice wants to send Bob a message. Alice encrypts some information using Bob's public key; Bob decrypts the ciphertext using his private key. This method could be also used to prove who sent a message; Alice, for example, could encrypt some plaintext with her private key; when Bob decrypts using Alice's public key, he knows that Alice sent the message and Alice cannot deny having sent the message (non-repudiation).
Public-key cryptography algorithms that are in use today for key exchange or digital signatures include:
RSA: The first, and still most common, PKC implementation, named for the three MIT mathematicians who developed it — Ronald Rivest, Adi Shamir, and Leonard Adleman. RSA today is used in hundreds of software products and can be used for key exchange, digital signatures, or encryption of small blocks of data. RSA uses a variable size encryption block and a variable size key. The key-pair is derived from a very large number, n, that is the product of two prime numbers chosen according to special rules; these primes may be 100 or more digits in length each, yielding an n with roughly twice as many digits as the prime factors. The public key information includes n and a derivative of one of the factors of n; an attacker cannot determine the prime factors of n (and, therefore, the private key) from this information alone and that is what makes the RSA algorithm so secure. (Some descriptions of PKC erroneously state that RSA's safety is due to the difficulty in factoring large prime numbers. In fact, large prime numbers, like small prime numbers, only have two factors!) The ability for computers to factor large numbers, and therefore attack schemes such as RSA, is rapidly improving and systems today can find the prime factors of numbers with more than 200 digits. Nevertheless, if a large number is created from two prime factors that are roughly the same size, there is no known factorization algorithm that will solve the problem in a reasonable amount of time; a 2005 test to factor a 200-digit number took 1.5 years and over 50 years of compute time (see the Wikipedia article on integer factorization.) Regardless, one presumed protection of RSA is that users can easily increase the key size to always stay ahead of the computer processing curve. As an aside, the patent for RSA expired in September 2000 which does not appear to have affected RSA's popularity one way or the other. A detailed example of RSA is presented below in Section 5.3.
Diffie-Hellman: After the RSA algorithm was published, Diffie and Hellman came up with their own algorithm. D-H is used for secret-key key exchange only, and not for authentication or digital signatures. More detail about Diffie-Hellman can be found below in Section 5.2.
Digital Signature Algorithm (DSA): The algorithm specified in NIST's Digital Signature Standard (DSS), provides digital signature capability for the authentication of messages.
ElGamal: Designed by Taher Elgamal, a PKC system similar to Diffie-Hellman and used for key exchange.
Elliptic Curve Cryptography (ECC): A PKC algorithm based upon elliptic curves. ECC can offer levels of security with small keys comparable to RSA and other PKC methods. It was designed for devices with limited compute power and/or memory, such as smartcards and PDAs. More detail about ECC can be found below in Section 5.8. Other references include "The Importance of ECC" Web page and the "Online Elliptic Curve Cryptography Tutorial", both from Certicom. See also RFC 6090 for a review of fundamental ECC algorithms.
Public-Key Cryptography Standards (PKCS): A set of interoperable standards and guidelines for public-key cryptography, designed by RSA Data Security Inc.
- PKCS #1: RSA Cryptography Standard (Also RFC 3447)
- PKCS #2: Incorporated into PKCS #1.
- PKCS #3: Diffie-Hellman Key-Agreement Standard
- PKCS #4: Incorporated into PKCS #1.
- PKCS #5: Password-Based Cryptography Standard (PKCS #5 V2.0 is also RFC 2898)
- PKCS #6: Extended-Certificate Syntax Standard (being phased out in favor of X.509v3)
- PKCS #7: Cryptographic Message Syntax Standard (Also RFC 2315)
- PKCS #8: Private-Key Information Syntax Standard (Also RFC 5208)
- PKCS #9: Selected Attribute Types (Also RFC 2985)
- PKCS #10: Certification Request Syntax Standard (Also RFC 2986)
- PKCS #11: Cryptographic Token Interface Standard
- PKCS #12: Personal Information Exchange Syntax Standard
- PKCS #13: Elliptic Curve Cryptography Standard
- PKCS #14: Pseudorandom Number Generation Standard is no longer available
- PKCS #15: Cryptographic Token Information Format Standard
Cramer-Shoup: A public-key cryptosystem proposed by R. Cramer and V. Shoup of IBM in 1998.
Key Exchange Algorithm (KEA): A variation on Diffie-Hellman; proposed as the key exchange method for Capstone.
LUC: A public-key cryptosystem designed by P.J. Smith and based on Lucas sequences. Can be used for encryption and signatures, using integer factoring.
For additional information on PKC algorithms, see "Public-Key Encryption", Chapter 8 in Handbook of Applied Cryptography, by A. Menezes, P. van Oorschot, and S. Vanstone (CRC Press, 1996).
A digression: Who invented PKC? I tried to be careful in the first paragraph of this section to state that Diffie and Hellman "first described publicly" a PKC scheme. Although I have categorized PKC as a two-key system, that has been merely for convenience; the real criteria for a PKC scheme is that it allows two parties to exchange a secret even though the communication with the shared secret might be overheard. There seems to be no question that Diffie and Hellman were first to publish; their method is described in the classic paper, "New Directions in Cryptography," published in the November 1976 issue of IEEE Transactions on Information Theory. As shown below, Diffie-Hellman uses the idea that finding logarithms is relatively harder than exponentiation. And, indeed, it is the precursor to modern PKC which does employ two keys. Rivest, Shamir, and Adleman described an implementation that extended this idea in their paper "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems," published in the February 1978 issue of the Communications of the ACM (CACM). Their method, of course, is based upon the relative ease of finding the product of two large prime numbers compared to finding the prime factors of a large number.
Some sources, though, credit Ralph Merkle with first describing a system that allows two parties to share a secret although it was not a two-key system, per se. A Merkle Puzzle works where Alice creates a large number of encrypted keys, sends them all to Bob so that Bob chooses one at random and then lets Alice know which he has selected. An eavesdropper will see all of the keys but can't learn which key Bob has selected (because he has encrypted the response with the chosen key). In this case, Eve's effort to break in is the square of the effort of Bob to choose a key. While this difference may be small it is often sufficient. Merkle apparently took a computer science course at UC Berkeley in 1974 and described his method, but had difficulty making people understand it; frustrated, he dropped the course. Meanwhile, he submitted the paper "Secure Communication Over Insecure Channels" which was published in the CACM in April 1978; Rivest et al.'s paper even makes reference to it. Merkle's method certainly wasn't published first, but did he have the idea first?
An interesting question, maybe, but who really knows? For some time, it was a quiet secret that a team at the UK's Government Communications Headquarters (GCHQ) had first developed PKC in the early 1970s. Because of the nature of the work, GCHQ kept the original memos classified. In 1997, however, the GCHQ changed their posture when they realized that there was nothing to gain by continued silence. Documents show that a GCHQ mathematician named James Ellis started research into the key distribution problem in 1969 and that by 1975, Ellis, Clifford Cocks, and Malcolm Williamson had worked out all of the fundamental details of PKC, yet couldn't talk about their work. (They were, of course, barred from challenging the RSA patent!) After more than 20 years, Ellis, Cocks, and Williamson have begun to get their due credit.
And the National Security Agency (NSA) claims to have knowledge of this type of algorithm as early as 1966 but there is no supporting documentation... yet. So this really was a digression...
Hash functions, also called message digests and one-way encryption, are algorithms that, in some sense, use no key (Figure 1C). Instead, a fixed-length hash value is computed based upon the plaintext that makes it impossible for either the contents or length of the plaintext to be recovered. Hash algorithms are typically used to provide a digital fingerprint of a file's contents, often used to ensure that the file has not been altered by an intruder or virus. Hash functions are also commonly employed by many operating systems to encrypt passwords. Hash functions, then, provide a measure of the integrity of a file.
Hash algorithms that are in common use today include:
Message Digest (MD) algorithms: A series of byte-oriented algorithms that produce a 128-bit hash value from an arbitrary-length message.
MD5 (RFC 1321): Also developed by Rivest after potential weaknesses were reported in MD4; this scheme is similar to MD4 but is slower because more manipulation is made to the original data. MD5 has been implemented in a large number of products although several weaknesses in the algorithm were demonstrated by German cryptographer Hans Dobbertin in 1996 ("Cryptanalysis of MD5 Compress").
Secure Hash Algorithm (SHA): Algorithm for NIST's Secure Hash Standard (SHS). SHA-1 produces a 160-bit hash value and was originally published as FIPS 180-1 and RFC 3174. FIPS 180-2 (aka SHA-2) describes five algorithms in the SHS: SHA-1 plus SHA-224, SHA-256, SHA-384, and SHA-512 which can produce hash values that are 224, 256, 384, or 512 bits in length, respectively. SHA-224, -256, -384, and -512 are also described in RFC 4634.
RIPEMD: A series of message digests that initially came from the RIPE (RACE Integrity Primitives Evaluation) project. RIPEMD-160 was designed by Hans Dobbertin, Antoon Bosselaers, and Bart Preneel, and optimized for 32-bit processors to replace the then-current 128-bit hash functions. Other versions include RIPEMD-256, RIPEMD-320, and RIPEMD-128.
HAVAL (HAsh of VAriable Length): Designed by Y. Zheng, J. Pieprzyk and J. Seberry, a hash algorithm with many levels of security. HAVAL can create hash values that are 128, 160, 192, 224, or 256 bits in length.
Whirlpool: A relatively new hash function, designed by V. Rijmen and P.S.L.M. Barreto. Whirlpool operates on messages less than 2256 bits in length, and produces a message digest of 512 bits. The design of this has function is very different than that of MD5 and SHA-1, making it immune to the same attacks as on those hashes (see below).
Tiger: Designed by Ross Anderson and Eli Biham, Tiger is designed to be secure, run efficiently on 64-bit processors, and easily replace MD4, MD5, SHA and SHA-1 in other applications. Tiger/192 produces a 192-bit output and is compatible with 64-bit architectures; Tiger/128 and Tiger/160 produce a hash of length 128 and 160 bits, respectively, to provide compatibility with the other hash functions mentioned above.
(Readers might be interested in HashCalc, a Windows-based program that calculates hash values using a dozen algorithms, including MD5, SHA-1 and several variants, RIPEMD-160, and Tiger. Command line utilities that calculate hash values include sha_verify by Dan Mares [Windows; supports MD5, SHA-1, SHA-2] and md5deep [cross-platform; supports MD5, SHA-1, SHA-256, Tiger, and Whirlpool].)
Hash functions are sometimes misunderstood and some sources claim that no two files can have the same hash value. This is, in fact, not correct. Consider a hash function that provides a 128-bit hash value. There are, obviously, 2128 possible hash values. But there are a lot more than 2128possible files. Therefore, there have to be multiple files — in fact, there have to be an infinite number of files! — that can have the same 128-bit hash value.
The difficulty is finding two files with the same hash! What is, indeed, very hard to do is to try to create a file that has a given hash value so as to force a hash value collision — which is the reason that hash functions are used extensively for information security and computer forensics applications. Alas, researchers in 2004 found that practical collision attacks could be launched on MD5, SHA-1, and other hash algorithms. Readers interested in this problem should read the following:
- AccessData. (2006, April). MD5 Collisions: The Effect on Computer Forensics. AccessData White Paper.
- Burr, W. (2006, March/April). Cryptographic hash standards: Where do we go from here? IEEE Security & Privacy, 4(2), 88-91.
- Dwyer, D. (2009, June 3). SHA-1 Collision Attacks Now 252. SecureWorks Research blog.
- Gutman, P., Naccache, D., & Palmer, C.C. (2005, May/June). When hashes collide. IEEE Security & Privacy, 3(3), 68-71.
- Klima, V. (March 2005). Finding MD5 Collisions - a Toy For a Notebook.
- Lee, R. (2009, January 7). Law Is Not A Science: Admissibility of Computer Evidence and MD5 Hashes. SANS Computer Forensics blog.
- Thompson, E. (2005, February). MD5 collisions and the impact on computer forensics. Digital Investigation, 2(1), 36-40.
- Wang, X., Feng, D., Lai, X., & Yu, H. (2004, August). Collisions for Hash Functions MD4, MD5, HAVAL-128 and RIPEMD.
- Wang, X., Yin, Y.L., & Yu, H. (2005, February 13). Collision Search Attacks on SHA1.
Readers are also referred to the Eindhoven University of Technology HashClash Project Web site. An excellent overview of the situation with hash collisions (circa 2005) can be found in RFC 4270 (by P. Hoffman and B. Schneier, November 2005). And for additional information on hash functions, see David Hopwood's MessageDigest Algorithms page.
At this time, there is no obvious successor to MD5 and SHA-1 that could be put into use quickly; there are so many products using these hash functions that it could take many years to flush out all use of 128- and 160-bit hashes. That said, NIST announced in 2007 their Cryptographic Hash Algorithm Competition to find the next-generation secure hashing method. Dubbed SHA-3, this new scheme will augment FIPS 180-2. A list of submissions can be found at The SHA-3 Zoo. The SHA-3 standard may not be available until 2011 or 2012.
Certain extensions of hash functions are used for a variety of information security and digital forensics applications, such as:
- Hash libraries are sets of hash values corresponding to known files. A hash library of known good files, for example, might be a set of files known to be a part of an operating system, while a hash library of known bad files might be of a set of known child pornographic images.
- Rolling hashes refer to a set of hash values that are computed based upon a fixed-length "sliding window" through the input. As an example, a hash value might be computed on bytes 1-10 of a file, then on bytes 2-11, 3-12, 4-13, etc.
- Fuzzy hashes are an area of intense research and represent hash values that represent two inputs that are similar. Fuzzy hashes are used to detect documents, images, or other files that are close to each other with respect to content. See "Fuzzy Hashing" (PDF | PPT) by Jesse Kornblum for a good treatment of this topic.
So, why are there so many different types of cryptographic schemes? Why can't we do everything we need with just one?
The answer is that each scheme is optimized for some specific application(s). Hash functions, for example, are well-suited for ensuring data integrity because any change made to the contents of a message will result in the receiver calculating a different hash value than the one placed in the transmission by the sender. Since it is highly unlikely that two different messages will yield the same hash value, data integrity is ensured to a high degree of confidence.
Secret key cryptography, on the other hand, is ideally suited to encrypting messages, thus providing privacy and confidentiality. The sender can generate a session key on a per-message basis to encrypt the message; the receiver, of course, needs the same session key to decrypt the message.
Key exchange, of course, is a key application of public-key cryptography (no pun intended). Asymmetric schemes can also be used for non-repudiation and user authentication; if the receiver can obtain the session key encrypted with the sender's private key, then only this sender could have sent the message. Public-key cryptography could, theoretically, also be used to encrypt messages although this is rarely done because secret-key cryptography operates about 1000 times faster than public-key cryptography.
Figure 2 puts all of this together and shows how a hybrid cryptographic scheme combines all of these functions to form a secure transmission comprising digital signature and digital envelope. In this example, the sender of the message is Alice and the receiver is Bob.
A digital envelope comprises an encrypted message and an encrypted session key. Alice uses secret key cryptography to encrypt her message using the session key, which she generates at random with each session. Alice then encrypts the session key using Bob's public key. The encrypted message and encrypted session key together form the digital envelope. Upon receipt, Bob recovers the session secret key using his private key and then decrypts the encrypted message.
The digital signature is formed in two steps. First, Alice computes the hash value of her message; next, she encrypts the hash value with her private key. Upon receipt of the digital signature, Bob recovers the hash value calculated by Alice by decrypting the digital signature with Alice's public key. Bob can then apply the hash function to Alice's original message, which he has already decrypted (see previous paragraph). If the resultant hash value is not the same as the value supplied by Alice, then Bob knows that the message has been altered; if the hash values are the same, Bob should believe that the message he received is identical to the one that Alice sent.
This scheme also provides nonrepudiation since it proves that Alice sent the message; if the hash value recovered by Bob using Alice's public key proves that the message has not been altered, then only Alice could have created the digital signature. Bob also has proof that he is the intended receiver; if he can correctly decrypt the message, then he must have correctly decrypted the session key meaning that his is the correct private key.
In a recent article in the industry literature (circa 9/98), a writer made the claim that 56-bit keys do not provide as sufficient protection for DES today as they did in 1975 because computers are 1000 times faster today than in 1975. Therefore, the writer went on, we should be using 56,000-bit keys today instead of 56-bit keys to provide adequate protection. The conclusion was then drawn that because 56,000-bit keys are infeasible (true), we should accept the fact that we have to live with weak cryptography (false!). The major error here is that the writer did not take into account that the number of possible key values double whenever a single bit is added to the key length; thus, a 57-bit key has twice as many values as a 56-bit key (because 257 is two times 256). In fact, a 66-bit key would have 1024 times the possible values as a 56-bit key.
But this does bring up the issue, what is the precise significance of key length as it affects the level of protection?
In cryptography, size does matter. The larger the key, the harder it is to crack a block of encrypted data. The reason that large keys offer more protection is almost obvious; computers have made it easier to attack ciphertext by using brute force methods rather than by attacking the mathematics (which are generally well-known anyway). With a brute force attack, the attacker merely generates every possible key and applies it to the ciphertext. Any resulting plaintext that makes sense offers a candidate for a legitimate key. This was the basis, of course, of the EFF's attack on DES.
Until the mid-1990s or so, brute force attacks were beyond the capabilities of computers that were within the budget of the attacker community. Today, however, significant compute power is commonly available and accessible. General purpose computers such as PCs are already being used for brute force attacks. For serious attackers with money to spend, such as some large companies or governments, Field Programmable Gate Array (FPGA) or Application-Specific Integrated Circuits (ASIC) technology offers the ability to build specialized chips that can provide even faster and cheaper solutions than a PC. Consider that an AT&T ORCA chip (FPGA) costs $200 and can test 30 million DES keys per second, while a $10 ASIC chip can test 200 million DES keys per second (compared to a PC which might be able to test 40,000 keys per second).
The table below shows what DES key sizes are needed to protect data from attackers with different time and financial resources. This information is not merely academic; one of the basic tenets of any security system is to have an idea of what you are protecting and from who are you protecting it! The table clearly shows that a 40-bit key is essentially worthless today against even the most unsophisticated attacker. On the other hand, 56-bit keys are fairly strong unless you might be subject to some pretty serious corporate or government espionage. But note that even 56-bit keys are declining in their value and that the times in the table (1995 data) are worst cases.
TABLE 1. Minimum Key Lengths for Symmetric Ciphers.
Type of Attacker Budget Tool Time and Cost
Per Key Recovered
Key Length Needed
40 bits 56 bits Pedestrian Hacker Tiny Scavanged
1 week Infeasible 45 $400 FPGA 5 hours
50 Small Business $10,000 FPGA 12 minutes
55 Corporate Department $300K FPGA 24 seconds
60 ASIC 0.18 seconds
Big Company $10M FPGA 7 seconds
70 ASIC 0.005 seconds
Intelligence Agency $300M ASIC 0.0002 seconds
So, how big is big enough? DES, invented in 1975, is still in use today, nearly 25 years later. If we take that to be a design criteria (i.e., a 20-plus year lifetime) and we believe Moore's Law ("computing power doubles every 18 months"), then a key size extension of 14 bits (i.e., a factor of more than 16,000) should be adequate. The 1975 DES proposal suggested 56-bit keys; by 1995, a 70-bit key would have been required to offer equal protection and an 85-bit key will be necessary by 2015.
The discussion above suggests that a 128- or 256-bit key for SKC will suffice for some time because that key length keeps us ahead of the brute force capabilities of the attackers. While a large key is good, a huge key may not always be better. That is, many public-key cryptosystems use 1024- or 2048-bit keys; expanding the key to 4096 bits probably doesn't add any protection at this time but it does add significantly to processing time.
The most effective large-number factoring methods today use a mathematical Number Field Sieve to find a certain number of relationships and then uses a matrix operation to solve a linear equation to produce the two prime factors. The sieve step actually involves a large number of operations of operations that can be performed in parallel; solving the linear equation, however, requires a supercomputer. Indeed, finding the solution to the RSA-140 challenge in February 1999 — factoring a 140-digit (465-bit) prime number — required 200 computers across the Internet about 4 weeks for the first step and a Cray computer 100 hours and 810 MB of memory to do the second step.
In early 1999, Shamir (of RSA fame) described a new machine that could increase factorization speed by 2-3 orders of magnitude. Although no detailed plans were provided nor is one known to have b