Размер ключа в криптографии: безопасность и стойкость

Number of bits in a key used by a cryptographic algorithm
In cryptography, key size or key length refers to the number of bits in a key used by a cryptographic algorithm (such as a cipher). Key length defines the upper bound on an algorithm's security (i. e. a logarithmic measure of the fastest known attack against an algorithm), because the security of all algorithms can be violated by brute force attacks. Ideally, the lower bound on an algorithm's security is by design equal to the key length (that is, the algorithm's design does not detract from the degree of security inherent in the key length). Most symmetric key algorithms are designed to have security equal to their key length. However, after design, a new attack might be discovered. For instance, Triple DES was designed to have a 168 bit key, but an attack of complexity 2112 is now known (i. e. Triple DES now only has 112 bits of security, and of the 168 bits in the key the attack has rendered 56 'ineffective' towards security). Nevertheless, as long as the security (understood as "the amount of effort it would take to gain access") is sufficient for a particular application, then it does not matter if key length and security coincide. This is important for asymmetric key algorithms, because no such algorithm is known to satisfy this property; elliptic curve cryptography comes the closest with an effective security of roughly half its key length.

Keys are used to control the operation of a cipher so that only the correct key can convert encrypted text (ciphertext) to plaintext. All commonly used ciphers are based on publicly known algorithms or are open source and so it is only the difficulty of obtaining the key that determines security of the system, provided that there is no analytic attack (i. e. a "structural weakness" in the algorithms or protocols used), and assuming that the key is not otherwise available (such as via theft, extortion, or compromise of computer systems). The widely accepted notion that the security of the system should depend on the key alone has been explicitly formulated by Auguste Kerckhoffs (in the 1880s) and Claude Shannon (in the 1940s); the statements are known as Kerckhoffs' principle and Shannon's Maxim respectively. A key should, therefore, be large enough that a brute force attack (possible against any encryption algorithm) is infeasible – i. e. would take too long and/or would take too much memory to execute. Shannon's work on information theory showed that to achieve so called 'perfect secrecy', the key length must be at least as large as the message and only used once (this algorithm is called the one time pad). In light of this, and the practical difficulty of managing such long keys, modern cryptographic practice has discarded the notion of perfect secrecy as a requirement for encryption, and instead focuses on computational security, under which the computational requirements of breaking an encrypted text must be infeasible for an attacker.

Encryption systems are often grouped into families. Common families include symmetric systems (e. g. AES) and asymmetric systems (e. g. RSA and Elliptic curve cryptography [ECC]). They may be grouped according to the central algorithm used (e. g. ECC and Feistel ciphers). Because each of these has a different level of cryptographic complexity, it is usual to have different key sizes for the same level of security, depending upon the algorithm used. For example, the security available with a 1024 bit key using asymmetric RSA is considered approximately equal in security to an 80 bit key in a symmetric algorithm. The factored number was of a special form; the special number field sieve cannot be used on RSA keys. The computation is roughly equivalent to breaking a 700 bit RSA key. However, this might be an advance warning that 1024 bit RSA keys used in secure online commerce should be deprecated, since they may become breakable in the foreseeable future. Cryptography professor Arjen Lenstra observed that "Last time, it took nine years for us to generalize from a special to a nonspecial, hard to factor number" and when asked whether 1024 bit RSA keys are dead, said: "The answer to that question is an unqualified yes." The 2015 Logjam attack revealed additional dangers in using Diffie Hellman key exchange when only one or a few common 1024 bit or smaller prime moduli are in use. This practice, somewhat common at the time, allows large amounts of communications to be compromised at the expense of attacking a small number of primes.

Even if a symmetric cipher is currently unbreakable by exploiting structural weaknesses in its algorithm, it may be possible to run through the entire space of keys in what is known as a brute force attack. Because longer symmetric keys require exponentially more work to brute force search, a sufficiently long symmetric key makes this line of attack impractical. With a key of length n bits, there are 2n possible keys. This number grows very rapidly as n increases. The large number of operations (2128) required to try all possible 128 bit keys is widely considered out of reach for conventional digital computing techniques for the foreseeable future. However, a quantum computer capable of running Grover's algorithm would be able to search the possible keys more efficiently. If a suitably sized quantum computer would reduce a 128 bit key down to 64 bit security, roughly a DES equivalent. This is one of the reasons why AES supports key lengths of 256 bits and longer.

IBM's Lucifer cipher was selected in 1974 as the base for what would become the Data Encryption Standard. Lucifer's key length was reduced from 128 bits to 56 bits, which the NSA and NIST argued was sufficient for non governmental protection at the time. The NSA has major computing resources and a large budget; some cryptographers including Whitfield Diffie and Martin Hellman complained that this made the cipher so weak that NSA computers would be able to break a DES key in a day through brute force parallel computing. The NSA disputed this, claiming that brute forcing DES would take them "something like 91 years". However, by the late 90s, it became clear that DES could be cracked in a few days' time frame with custom built hardware such as could be purchased by a large corporation or government. The book Cracking DES (O'Reilly and Associates) tells of the successful ability in 1998 to break 56 bit DES by a brute force attack mounted by a cyber civil rights group with limited resources; see EFF DES cracker. Even before that demonstration, 56 bits was considered insufficient length for symmetric algorithm keys for general use. Because of this, DES was replaced in most security applications by Triple DES, which has 112 bits of security when using 168 bit keys (triple key). Since 2015, NIST guidance says that "the use of keys that provide less than 112 bits of security strength for key agreement is now disallowed." NIST approved symmetric encryption algorithms include three key Triple DES, and AES. Approvals for two key Triple DES and Skipjack were withdrawn in 2015; the NSA's Skipjack algorithm used in its Fortezza program employs 80 bit keys.

The effectiveness of public key cryptosystems depends on the intractability (computational and theoretical) of certain mathematical problems such as integer factorization. These problems are time consuming to solve, but usually faster than trying all possible keys by brute force. Thus, asymmetric keys must be longer for equivalent resistance to attack than symmetric algorithm keys. The most common methods are assumed to be weak against sufficiently powerful quantum computers in the future. Since 2015, NIST recommends a minimum of 2048 bit keys for RSA, an update to the widely accepted recommendation of a 1024 bit minimum since at least 2002. 1024 bit RSA keys are equivalent in strength to 80 bit symmetric keys, 2048 bit RSA keys to 112 bit symmetric keys, 3072 bit RSA keys to 128 bit symmetric keys, and 15360 bit RSA keys to 256 bit symmetric keys. In 2003, RSA Security claimed that 1024 bit keys were likely to become crackable sometime between 2006 and 2010, while 2048 bit keys are sufficient until 2030. as of 2020 the largest RSA key publicly known to be cracked is RSA 250 with 829 bits. The Finite Field Diffie Hellman algorithm has roughly the same key strength as RSA for the same key sizes. The work factor for breaking Diffie Hellman is based on the discrete logarithm problem, which is related to the integer factorization problem on which RSA's strength is based. Thus, a 2048 bit Diffie Hellman key has about the same strength as a 2048 bit RSA key. Elliptic curve cryptography (ECC) is an alternative set of asymmetric algorithms that is equivalently secure with shorter keys, requiring only approximately twice the bits as the equivalent symmetric algorithm. A 256 bit Elliptic curve Diffie–Hellman (ECDH) key has approximately the same safety factor as a 128 bit AES key. The NSA previously recommended 256 bit ECC for protecting classified information up to the SECRET level, and 384 bit for TOP SECRET; In 2015 it announced plans to transition to quantum resistant algorithms by 2024, and until then recommends 384 bit for all classified information.