CRYTOGRAPHY A-Z Introduction Algorithms Protocols & Standards References FAQs Links

Introduction to Cryptography

• Preface
• Basic Terminology
• Basic Cryptographic Algorithms
• Digital Signatures
• Cryptographic Hash Functions
• Cryptographic Random Number Generators
• Strength of Cryptographic Algorithms
• Cryptanalysis and Attacks on Cryptosystems

Preface

As we move into an information society, the technological means for global surveillance of millions of individual people are becoming available to major govenments. Cryptography has become one of the main tools for privacy, trust, access control, electronic payments, corporate security, and countless other fields.

Cryptography is no longer a military thing that should not be messed with. It is time to demystify cryptography and make full use of the advantages it provides for the modern society. In the following, basic terminology and the main methods of cryptography are presented. Any opinions and evaluations presented here are speculative, and the author cannot be held responsible for their correctness.

Basic Terminology

In cryptographic terminology, the message is called plaintext or cleartext. Encoding the contents of the message in such a way that hides its contents from outsiders is called encryption. The encrypted message is called the ciphertext. The process of retrieving the plaintext from the ciphertext is called decryption. Encryption and decryption usually make use of a key, and the coding method is such that decryption can be performed only by knowing the proper key.

Cryptography is the art or science of keeping messages secret. Cryptanalysis is the art of breaking ciphers, i.e. retrieving the plaintext without knowing the proper key. People who do cryptography are cryptographers, and practitioners of cryptanalysis are cryptanalysts.

Cryptography deals with all aspects of secure messaging, authentication, digital signatures, electronic money, and other applications. Cryptology is the branch of mathematics that studies the mathematical foundations of cryptographic methods.

Basic Cryptographic Algorithms

There are two classes of key-based algorithms, symmetric (or secret-key) and asymmetric (or public-key) algorithms. The difference is that symmetric algorithms use the same key for encryption and decryption (or the decryption key is easily derived from the encryption key), whereas asymmetric algorithms use a different key for encryption and decryption, and the decryption key cannot be derived from the encryption key.

Symmetric algorithms can be divided into stream ciphers and block ciphers. Stream ciphers can encrypt a single bit of plaintext at a time, whereas block ciphers take a number of bits (typically 64 bits in modern ciphers), and encrypt them as a single unit. Many symmetric ciphers are described on the algorithms page.

Asymmetric ciphers (also called public-key algorithms or generally public-key cryptography) permit the encryption key to be public (it can even be published in a newspaper), allowing anyone to encrypt with the key, whereas only the proper recipient (who knows the decryption key) can decrypt the message. The encryption key is also called the public key and the decryption key the private key or secret key.

Modern cryptographic algorithms cannot really be executed by humans. Strong cryptographic algorithms are designed to be executed by computers or specialized hardware devices. In most applications, cryptography is done in computer software.

Generally, symmetric algorithms are much faster to execute on a computer than asymmetric ones. In practice they are often used together, so that a public-key algorithm is used to encrypt a randomly generated encryption key, and the random key is used to encrypt the actual message using a symmetric algorithm.

Many good cryptographic algorithms are widely and publicly available in any major bookstore, scientific library, or patent office, on the Internet. Well-known symmetric functions include DES and IDEA. RSA is probably the best known asymmetric algorithm. The books page lists several good textbooks on cryptography and related topics.

Digital Signatures

Digital signatures are used to verify that a message really comes from the claimed sender (assuming only the sender knows the secret key corresponding to his/her public key). They can also be used to timestamp documents: a trusted party signs the document and its timestamp with his/her secret key, thus testifying that the document existed at the stated time.

Digital signatures can also be used to testify (or certify) that a public key belongs to a particular person. This is done by signing the combination of the key and the information about its owner by a trusted key. The reason for trusting that key may again be that it was signed by another trusted key. Eventually some key must be a root of the trust hierarchy (that is, it is not trusted because it was signed by somebody, but because you believe a priori that the key can be trusted). In a centralized key infrastructure there are very few roots in the trust network (e.g., trusted government agencies; such roots are also called certification authorities). In a distributed infrastructure there need not be any universally accepted roots, and each party may have different trusted roots (such of the party's own key and any keys signed by it). This is the web of trust concept used e.g. in PGP.

A digital signature of an arbitrary document is typically created by computing a message digest from the document, and concatenating it with information about the signer, a timestamp, etc. The resulting string is then encrypted using the private key of the signer using a suitable algorithm. The resulting encrypted block of bits is the signature. It is often distributed together with information about the public key that was used to sign it. To verify a signature, the recipient first determines whether it trusts that the key belongs to the person it is supposed to belong to (using the web of trust or a priori knowledge), and then decrypts the signature using the public key of the person. If the signature decrypts properly and the information matches that of the message (proper message digest etc.), the signature is accepted as valid.

Several methods for making and verifying digital signatures are freely available. The most widely known algorithm is RSA.

Cryptographic Hash Functions

Cryptographic hash functions typically produce hash values of 128 or more bits. This number is vastly larger than the number of different messages likely to ever be exchanged in the world.

Many good cryptographic hash functions are freely available. Well-known ones include MD5 and SHA.

Cryptographic Random Number Generators

In the optimal case, random numbers are based on true physical sources of randomness that cannot be predicted. Such sources may include the noise from a semiconductor device, the least significant bits of an audio input, or the intervals between device interrupts or user keystrokes. The noise obtained from a physical source is then "distilled" by a cryptographic hash function to make every bit depend on every other bit. Quite often a large pool (several thousand bits) is used to contain randomness, and every bit of the pool is made to depend on every bit of input noise and every other bit of the pool in a cryptographically strong way.

When true physical randomness is not available, pseudorandom numbers must be used. This situation is undesirable, but often arises on general purpose computers. It is always desirable to obtain some environmental noise - even from device latencies, resource utilization statistics, network statistics, keyboard interrupts, or whatever. The point is that the data must be unpredictable for any external observer; to achieve this, the random pool must contain at least 128 bits of true entropy.

Cryptographic pseudorandom generators typically have a large pool ("seed value") containing randomness. Bits are returned from this pool by taking data from the pool, optionally running the data through a cryptographic hash function to avoid revealing the contents of the pool. When more bits are needed, the pool is stirred by encrypting its contents by a suitable cipher with a random key (that may be taken from an unreturned part of the pool) in a mode which makes every bit of the pool depend on every other bit of the pool. New environmental noise should be mixed into the pool before stirring to make predicting previous or future values even more impossible.

Even though cryptographically strong random number generators are not very difficult to built if designed properly, they are often overlooked. The importance of the random number generator must thus be emphasized - if done badly, it will easily become the weakest point of the system.

Several examples of cryptographic random number generators are publicly available.

Strength of Cryptographic Algorithms

In theory, any cryptographic method with a key can be broken by trying all possible keys in sequence. If using brute force to try all keys is the only option, the required computing power increases exponentially with the length of the key. A 32 bit key takes 2^32 (about 10^9) steps. This is something any amateur can do on his/her home computer. A system with 40 bit keys (e.g. US-exportable version of RC4) takes 2^40 steps - this kind of computing power is available in most universities and even smallish companies. A system with 56 bit keys (such as DES) takes a substantial effort, but is quite easily breakable with special hardware. The cost of the special hardware is substantial but easily within reach of organized criminals, major companies, and governments. Keys with 64 bits are probably breakable now by major governments, and will be within reach of organized criminals, major companies, and lesser governments in a few years. Keys with 80 bits may become breakable in future. Keys with 128 bits will probably remain unbreakable by brute force for the foreseeable future. Even larger keys are of course possible. However, key length is not the only relevant issue. Many ciphers can be broken without trying all possible keys. In general, it is very difficult to design ciphers that could not be broken more effectively using other methods. Designing your own ciphers may be fun, but it is not recommended in real applications unless you are a true expert and know exactly what you are doing.

One should generally be very wary of unpublished or secret algorithms. Quite often the designer is then not sure of the security of the algorithm, or its security depends on the secrecy of the algorithm. Generally, no algorithm that depends on the secrecy of the algorithm is secure. Particularly in software, anyone can hire someone to disassemble and reverse-engineer the algorithm. Experience has shown that a vast majority of secret algorithms that have become public knowledge later have been pitifully weak in reality.

The key lengths used in public-key cryptography are usually much longer than those used in symmetric ciphers. There the problem is not that of guessing the right key, but deriving the matching secret key from the public key. In the case of RSA, this is equivalent to factoring a large integer that has two large prime factors. In the case of some other cryptosystems it is equivalent to computing the discrete logarithm modulo a large integer (which is believed to be roughly comparable to factoring). Other cryptosystems are based on yet other problems.

To give some idea of the complexity, for the RSA cryptosystem, a 256 bit modulus is easily factored by ordinary people. 384 bit keys can be broken by university research groups or companies. 512 bits is within reach of major governments. Keys with 768 bits are probably not secure in the long term. Keys with 1024 bits and more should be safe for now unless major algorithmic advances are made in factoring; keys of 2048 bits are considered by many to be secure for decades. It should be emphasized that the strength of a cryptographic system is usually equal to its weakest point. No aspect of the system design should be overlooked, from the choice algorithms to the key distribution and usage policies.

Cryptanalysis and Attacks on Cryptosystems

• Ciphertext-only attack: This is the situation where the attacker does not know anything about the contents of the message, and must work from ciphertext only. In practice it is quite often possible to make guesses about the plaintext, as many types of messages have fixed format headers. Even ordinary letters and documents begin in a very predictable way. It may also be possible to guess that some ciphertext block contains a common word.

• Known-plaintext attack: The attacker knows or can guess the plaintext for some parts of the ciphertext. The task is to decrypt the rest of the ciphertext blocks using this information. This may be done by determining the key used to encrypt the data, or via some shortcut.

• Chosen-plaintext attack: The attacker is able to have any text he likes encrypted with the unknown key. The task is to determine the key used for encryption. Some encryption methods, particularly RSA, are extremely vulnerable to chosen-plaintext attacks. When such algorithms are used, extreme care must be taken to design the entire system so that an attacker can never have chosen plaintext encrypted.

• Man-in-the-middle attack: This attack is relevant for cryptographic communication and key exchange protocols. The idea is that when two parties are exchanging keys for secure communications (e.g., using Diffie-Hellman), an adversary puts himhelf between the parties on the communication line. The adversary then performs a separate key exchange with each party. The parties will end up using a different key, each of which is known to the adversary. The adversary will then decrypt any communications with the proper key, and encrypt them with the other key for sending to the other party. The parties will think that they are communicating securely, but in fact the adversary is hearing everything.

  One way to prevent man-in-the-middle attacks is that both sides compute a cryptographic hash function of the key exchange (or at least the encryption keys), sign it using a digital signature algorithm, and send the signature to the other side. The recipient then verifies that the signature came from the desired other party, and that the hash in the signature matches that computed locally. This method is used e.g. in Photuris.

• Timing Attack: This very recent attack is based on repeatedly measuring the exact execution times of modular exponentiation operations. It is relevant to at least RSA, Diffie-Hellman, and Elliptic Curve methods. More information is available in the original paper.