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107-431: Most cryptographic applications require random numbers, for example: The "quality" of the randomness required for these applications varies. For example, creating a nonce in some protocols needs only uniqueness. On the other hand, the generation of a master key requires a higher quality, such as more entropy . And in the case of one-time pads , the information-theoretic guarantee of perfect secrecy only holds if

214-483: A + 1 b − a + 1 {\displaystyle F(k;a,b)={\frac {\lfloor k\rfloor -a+1}{b-a+1}}} on the distribution's support k ∈ [ a , b ] . {\textstyle k\in [a,b].} The problem of estimating the maximum N {\displaystyle N} of a discrete uniform distribution on the integer interval [ 1 , N ] {\displaystyle [1,N]} from

321-440: A , b ) = min ( max ( ⌊ k ⌋ − a + 1 b − a + 1 , 0 ) , 1 ) , {\displaystyle F(k;a,b)=\min \left(\max \left({\frac {\lfloor k\rfloor -a+1}{b-a+1}},0\right),1\right),} or simply F ( k ; a , b ) = ⌊ k ⌋ −

428-473: A chosen-plaintext attack , Eve may choose a plaintext and learn its corresponding ciphertext (perhaps many times); an example is gardening , used by the British during WWII. In a chosen-ciphertext attack , Eve may be able to choose ciphertexts and learn their corresponding plaintexts. Finally in a man-in-the-middle attack Eve gets in between Alice (the sender) and Bob (the recipient), accesses and modifies

535-428: A classical cipher (and some modern ciphers) will reveal statistical information about the plaintext, and that information can often be used to break the cipher. After the discovery of frequency analysis , nearly all such ciphers could be broken by an informed attacker. Such classical ciphers still enjoy popularity today, though mostly as puzzles (see cryptogram ). The Arab mathematician and polymath Al-Kindi wrote

642-499: A simple algorithm can remove a considerable amount of the bias in any bit stream, which should be applied to each bit stream before using any variation of the Santha–Vazirani design. CSPRNG designs are divided into two classes: "Practical" CSPRNG schemes not only include an CSPRNG algorithm, but also a way to initialize (" seed ") it while keeping the seed secret. A number of such schemes have been defined, including: Obviously,

749-642: A $ 10 million payment from the NSA to do so. On October 23, 2017, Shaanan Cohney , Matthew Green , and Nadia Heninger , cryptographers at the University of Pennsylvania and Johns Hopkins University , released details of the DUHK (Don't Use Hard-coded Keys) attack on WPA2 where hardware vendors use a hardcoded seed key for the ANSI X9.31 RNG algorithm, stating "an attacker can brute-force encrypted data to discover

856-645: A PRNG that shows the random numbers not to be truly random. Second, for most PRNGs, when their state has been revealed, all past random numbers can be retrodicted, allowing an attacker to read all past messages, as well as future ones. CSPRNGs are designed explicitly to resist this type of cryptanalysis . In the asymptotic setting , a family of deterministic polynomial time computable functions G k : { 0 , 1 } k → { 0 , 1 } p ( k ) {\displaystyle G_{k}\colon \{0,1\}^{k}\to \{0,1\}^{p(k)}} for some polynomial p ,

963-638: A book on cryptography entitled Risalah fi Istikhraj al-Mu'amma ( Manuscript for the Deciphering Cryptographic Messages ), which described the first known use of frequency analysis cryptanalysis techniques. Language letter frequencies may offer little help for some extended historical encryption techniques such as homophonic cipher that tend to flatten the frequency distribution. For those ciphers, language letter group (or n-gram) frequencies may provide an attack. Essentially all ciphers remained vulnerable to cryptanalysis using

1070-430: A cryptographic hash function is computed, and only the resulting hash is digitally signed. Cryptographic hash functions are functions that take a variable-length input and return a fixed-length output, which can be used in, for example, a digital signature. For a hash function to be secure, it must be difficult to compute two inputs that hash to the same value ( collision resistance ) and to compute an input that hashes to

1177-457: A cryptographically secure PRNG, but the reverse is not true. CSPRNG requirements fall into two groups: For instance, if the PRNG under consideration produces output by computing bits of pi in sequence, starting from some unknown point in the binary expansion, it may well satisfy the next-bit test and thus be statistically random, as pi is conjectured to be a normal number . However, this algorithm

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1284-427: A discrete uniform distribution are not numbered in order but are recognizable or markable, one can instead estimate population size via a mark and recapture method. See rencontres numbers for an account of the probability distribution of the number of fixed points of a uniformly distributed random permutation . The family of uniform discrete distributions over ranges of integers with one or both bounds unknown has

1391-532: A fair six-sided die . The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of each given value is 1/6. If two dice were thrown and their values added, the possible sums would not have equal probability and so the distribution of sums of two dice rolls is not uniform. Although it is common to consider discrete uniform distributions over a contiguous range of integers, such as in this six-sided die example, one can define discrete uniform distributions over any finite set . For instance,

1498-631: A finite-dimensional sufficient statistic , namely the triple of the sample maximum, sample minimum, and sample size. Uniform discrete distributions over bounded integer ranges do not constitute an exponential family of distributions because their support varies with their parameters. For families of distributions in which their supports do not depend on their parameters, the Pitman–Koopman–Darmois theorem states that only exponential families have sufficient statistics of dimensions that are bounded as sample size increases. The uniform distribution

1605-704: A forward secure PRNG with block length p ( k ) − k {\displaystyle p(k)-k} by splitting its output into the next state and the actual output. This is done by setting G ( s ) = G 0 ( s ) ‖ G 1 ( s ) {\displaystyle G(s)=G_{0}(s)\Vert G_{1}(s)} , in which | G 0 ( s ) | = | s | = k {\displaystyle |G_{0}(s)|=|s|=k} and | G 1 ( s ) | = p ( k ) − k {\displaystyle |G_{1}(s)|=p(k)-k} ; then G

1712-516: A given output ( preimage resistance ). MD4 is a long-used hash function that is now broken; MD5 , a strengthened variant of MD4, is also widely used but broken in practice. The US National Security Agency developed the Secure Hash Algorithm series of MD5-like hash functions: SHA-0 was a flawed algorithm that the agency withdrew; SHA-1 is widely deployed and more secure than MD5, but cryptanalysts have identified attacks against it;

1819-486: A good cipher to maintain confidentiality under an attack. This fundamental principle was first explicitly stated in 1883 by Auguste Kerckhoffs and is generally called Kerckhoffs's Principle ; alternatively and more bluntly, it was restated by Claude Shannon , the inventor of information theory and the fundamentals of theoretical cryptography, as Shannon's Maxim —'the enemy knows the system'. Different physical devices and aids have been used to assist with ciphers. One of

1926-428: A hashed output that cannot be used to retrieve the original input data. Cryptographic hash functions are used to verify the authenticity of data retrieved from an untrusted source or to add a layer of security. Symmetric-key cryptosystems use the same key for encryption and decryption of a message, although a message or group of messages can have a different key than others. A significant disadvantage of symmetric ciphers

2033-440: A keystream (in place of a Pseudorandom number generator ) and applying an XOR operation to each bit of the plaintext with each bit of the keystream. Message authentication codes (MACs) are much like cryptographic hash functions , except that a secret key can be used to authenticate the hash value upon receipt; this additional complication blocks an attack scheme against bare digest algorithms , and so has been thought worth

2140-414: A more specific meaning: the replacement of a unit of plaintext (i.e., a meaningful word or phrase) with a code word (for example, "wallaby" replaces "attack at dawn"). A cypher, in contrast, is a scheme for changing or substituting an element below such a level (a letter, a syllable, or a pair of letters, etc.) to produce a cyphertext. Cryptanalysis is the term used for the study of methods for obtaining

2247-438: A polynomial time algorithm. A forward-secure PRNG with block length t ( k ) {\displaystyle t(k)} is a PRNG G k : { 0 , 1 } k → { 0 , 1 } k × { 0 , 1 } t ( k ) {\displaystyle G_{k}\colon \{0,1\}^{k}\to \{0,1\}^{k}\times \{0,1\}^{t(k)}} , where

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2354-700: A sample of k observations is commonly known as the German tank problem , following the practical application of this maximum estimation problem, during World War II , by Allied forces seeking to estimate German tank production. A uniformly minimum variance unbiased (UMVU) estimator for the distribution's maximum in terms of m, the sample maximum , and k, the sample size , is N ^ = k + 1 k m − 1 = m + m k − 1 {\displaystyle {\hat {N}}={\frac {k+1}{k}}m-1=m+{\frac {m}{k}}-1} . This can be seen as

2461-476: A secret key is used to process the message (or a hash of the message, or both), and one for verification , in which the matching public key is used with the message to check the validity of the signature. RSA and DSA are two of the most popular digital signature schemes. Digital signatures are central to the operation of public key infrastructures and many network security schemes (e.g., SSL/TLS , many VPNs , etc.). Public-key algorithms are most often based on

2568-520: A security perspective to develop a new standard to "significantly improve the robustness of NIST 's overall hash algorithm toolkit." Thus, a hash function design competition was meant to select a new U.S. national standard, to be called SHA-3 , by 2012. The competition ended on October 2, 2012, when the NIST announced that Keccak would be the new SHA-3 hash algorithm. Unlike block and stream ciphers that are invertible, cryptographic hash functions produce

2675-464: A shared secret key. In practice, asymmetric systems are used to first exchange a secret key, and then secure communication proceeds via a more efficient symmetric system using that key. Examples of asymmetric systems include Diffie–Hellman key exchange , RSA ( Rivest–Shamir–Adleman ), ECC ( Elliptic Curve Cryptography ), and Post-quantum cryptography . Secure symmetric algorithms include the commonly used AES ( Advanced Encryption Standard ) which replaced

2782-406: A story by Edgar Allan Poe . Until modern times, cryptography referred almost exclusively to "encryption", which is the process of converting ordinary information (called plaintext ) into an unintelligible form (called ciphertext ). Decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext. A cipher (or cypher) is a pair of algorithms that carry out

2889-606: A stream cipher. The Data Encryption Standard (DES) and the Advanced Encryption Standard (AES) are block cipher designs that have been designated cryptography standards by the US government (though DES's designation was finally withdrawn after the AES was adopted). Despite its deprecation as an official standard, DES (especially its still-approved and much more secure triple-DES variant) remains quite popular; it

2996-459: A true random number generator. When the maximum number of bits output from this PRNG is less than it, the expected security level is delivered and the output appears to be indistinguishable from a true random number generator. It is noted in the next revision that the claimed security strength for CTR_DRBG depends on limiting the total number of generate requests and the bits provided per generate request. The fourth and final PRNG in this standard

3103-422: A very simple case of maximum spacing estimation . This has a variance of so a standard deviation of approximately N k {\displaystyle {\tfrac {N}{k}}} , the population-average gap size between samples. The sample maximum m {\displaystyle m} itself is the maximum likelihood estimator for the population maximum, but it is biased. If samples from

3210-465: A wide variety of cryptanalytic attacks, and they can be classified in any of several ways. A common distinction turns on what Eve (an attacker) knows and what capabilities are available. In a ciphertext-only attack , Eve has access only to the ciphertext (good modern cryptosystems are usually effectively immune to ciphertext-only attacks). In a known-plaintext attack , Eve has access to a ciphertext and its corresponding plaintext (or to many such pairs). In

3317-446: A widely used tool in communications, computer networks , and computer security generally. Some modern cryptographic techniques can only keep their keys secret if certain mathematical problems are intractable , such as the integer factorization or the discrete logarithm problems, so there are deep connections with abstract mathematics . There are very few cryptosystems that are proven to be unconditionally secure. The one-time pad

Cryptographically secure pseudorandom number generator - Misplaced Pages Continue

3424-487: Is non-parametric . However, in the common case that its possible outcome values are the integers in an interval [ a , b ] {\textstyle [a,b]} , then a and b are parameters of the distribution and n = b − a + 1. {\textstyle n=b-a+1.} In these cases the cumulative distribution function (CDF) of the discrete uniform distribution can be expressed, for any k , as F ( k ;

3531-576: Is a pseudorandom number generator (PRNG, or PRG in some references), if it stretches the length of its input ( p ( k ) > k {\displaystyle p(k)>k} for any k ), and if its output is computationally indistinguishable from true randomness, i.e. for any probabilistic polynomial time algorithm A , which outputs 1 or 0 as a distinguisher, for some negligible function μ {\displaystyle \mu } . (The notation x ← X {\displaystyle x\gets X} means that x

3638-423: Is a forward secure PRNG with G 0 {\displaystyle G_{0}} as the next state and G 1 {\displaystyle G_{1}} as the pseudorandom output block of the current period. Santha and Vazirani proved that several bit streams with weak randomness can be combined to produce a higher-quality, quasi-random bit stream. Even earlier, John von Neumann proved that

3745-417: Is also active research examining the relationship between cryptographic problems and quantum physics . Just as the development of digital computers and electronics helped in cryptanalysis, it made possible much more complex ciphers. Furthermore, computers allowed for the encryption of any kind of data representable in any binary format, unlike classical ciphers which only encrypted written language texts; this

3852-497: Is also widely used but broken in practice. The US National Security Agency developed the Secure Hash Algorithm series of MD5-like hash functions: SHA-0 was a flawed algorithm that the agency withdrew; SHA-1 is widely deployed and more secure than MD5, but cryptanalysts have identified attacks against it; the SHA-2 family improves on SHA-1, but is vulnerable to clashes as of 2011; and the US standards authority thought it "prudent" from

3959-408: Is beyond the ability of any adversary. This means it must be shown that no efficient method (as opposed to the time-consuming brute force method) can be found to break the cipher. Since no such proof has been found to date, the one-time-pad remains the only theoretically unbreakable cipher. Although well-implemented one-time-pad encryption cannot be broken, traffic analysis is still possible. There are

4066-443: Is called cryptolinguistics . Cryptolingusitics is especially used in military intelligence applications for deciphering foreign communications. Before the modern era, cryptography focused on message confidentiality (i.e., encryption)—conversion of messages from a comprehensible form into an incomprehensible one and back again at the other end, rendering it unreadable by interceptors or eavesdroppers without secret knowledge (namely

4173-413: Is chosen uniformly at random from the set X .) There is an equivalent characterization: For any function family G k : { 0 , 1 } k → { 0 , 1 } p ( k ) {\displaystyle G_{k}\colon \{0,1\}^{k}\to \{0,1\}^{p(k)}} , G is a PRNG if and only if the next output bit of G cannot be predicted by

4280-648: Is claimed to have developed the Diffie–Hellman key exchange. Public-key cryptography is also used for implementing digital signature schemes. A digital signature is reminiscent of an ordinary signature; they both have the characteristic of being easy for a user to produce, but difficult for anyone else to forge . Digital signatures can also be permanently tied to the content of the message being signed; they cannot then be 'moved' from one document to another, for any attempt will be detectable. In digital signature schemes, there are two algorithms: one for signing , in which

4387-399: Is combined with the plaintext bit-by-bit or character-by-character, somewhat like the one-time pad . In a stream cipher, the output stream is created based on a hidden internal state that changes as the cipher operates. That internal state is initially set up using the secret key material. RC4 is a widely used stream cipher. Block ciphers can be used as stream ciphers by generating blocks of

Cryptographically secure pseudorandom number generator - Misplaced Pages Continue

4494-560: Is impossible; it is quite unusable in practice. The discrete logarithm problem is the basis for believing some other cryptosystems are secure, and again, there are related, less practical systems that are provably secure relative to the solvability or insolvability discrete log problem. As well as being aware of cryptographic history, cryptographic algorithm and system designers must also sensibly consider probable future developments while working on their designs. For instance, continuous improvements in computer processing power have increased

4601-535: Is named Dual EC DRBG . It has been shown to not be cryptographically secure and is believed to have a kleptographic NSA backdoor. A good reference is maintained by NIST . There are also standards for statistical testing of new CSPRNG designs: The Guardian and The New York Times reported in 2013 that the National Security Agency (NSA) inserted a backdoor into a pseudorandom number generator (PRNG) of NIST SP 800-90A , which allows

4708-458: Is not cryptographically secure; an attacker who determines which bit of pi is currently in use (i.e. the state of the algorithm) will be able to calculate all preceding bits as well. Most PRNGs are not suitable for use as CSPRNGs and will fail on both counts. First, while most PRNGs' outputs appear random to assorted statistical tests, they do not resist determined reverse engineering. Specialized statistical tests may be found specially tuned to such

4815-617: Is one, and was proven to be so by Claude Shannon. There are a few important algorithms that have been proven secure under certain assumptions. For example, the infeasibility of factoring extremely large integers is the basis for believing that RSA is secure, and some other systems, but even so, proof of unbreakability is unavailable since the underlying mathematical problem remains open. In practice, these are widely used, and are believed unbreakable in practice by most competent observers. There are systems similar to RSA, such as one by Michael O. Rabin that are provably secure provided factoring n = pq

4922-765: Is relatively recent, beginning in the mid-1970s. In the early 1970s IBM personnel designed the Data Encryption Standard (DES) algorithm that became the first federal government cryptography standard in the United States. In 1976 Whitfield Diffie and Martin Hellman published the Diffie–Hellman key exchange algorithm. In 1977 the RSA algorithm was published in Martin Gardner 's Scientific American column. Since then, cryptography has become

5029-454: Is the key management necessary to use them securely. Each distinct pair of communicating parties must, ideally, share a different key, and perhaps for each ciphertext exchanged as well. The number of keys required increases as the square of the number of network members, which very quickly requires complex key management schemes to keep them all consistent and secret. In a groundbreaking 1976 paper, Whitfield Diffie and Martin Hellman proposed

5136-827: Is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science , information security , electrical engineering , digital signal processing , physics, and others. Core concepts related to information security ( data confidentiality , data integrity , authentication , and non-repudiation ) are also central to cryptography. Practical applications of cryptography include electronic commerce , chip-based payment cards , digital currencies , computer passwords , and military communications . Cryptography prior to

5243-499: Is theoretically possible to break into a well-designed system, it is infeasible in actual practice to do so. Such schemes, if well designed, are therefore termed "computationally secure". Theoretical advances (e.g., improvements in integer factorization algorithms) and faster computing technology require these designs to be continually reevaluated and, if necessary, adapted. Information-theoretically secure schemes that provably cannot be broken even with unlimited computing power, such as

5350-439: Is to find some weakness or insecurity in a cryptographic scheme, thus permitting its subversion or evasion. It is a common misconception that every encryption method can be broken. In connection with his WWII work at Bell Labs , Claude Shannon proved that the one-time pad cipher is unbreakable, provided the key material is truly random , never reused, kept secret from all possible attackers, and of equal or greater length than

5457-434: Is typically the case that use of a quality cipher is very efficient (i.e., fast and requiring few resources, such as memory or CPU capability), while breaking it requires an effort many orders of magnitude larger, and vastly larger than that required for any classical cipher, making cryptanalysis so inefficient and impractical as to be effectively impossible. Symmetric-key cryptography refers to encryption methods in which both

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5564-419: Is used across a wide range of applications, from ATM encryption to e-mail privacy and secure remote access . Many other block ciphers have been designed and released, with considerable variation in quality. Many, even some designed by capable practitioners, have been thoroughly broken, such as FEAL . Stream ciphers, in contrast to the 'block' type, create an arbitrarily long stream of key material, which

5671-439: The r i {\displaystyle r_{i}} are chosen uniformly at random from { 0 , 1 } t ( k ) {\displaystyle \{0,1\}^{t(k)}} . Any PRNG G : { 0 , 1 } k → { 0 , 1 } p ( k ) {\displaystyle G\colon \{0,1\}^{k}\to \{0,1\}^{p(k)}} can be turned into

5778-442: The SHA-2 family improves on SHA-1, but is vulnerable to clashes as of 2011; and the US standards authority thought it "prudent" from a security perspective to develop a new standard to "significantly improve the robustness of NIST 's overall hash algorithm toolkit." Thus, a hash function design competition was meant to select a new U.S. national standard, to be called SHA-3 , by 2012. The competition ended on October 2, 2012, when

5885-557: The computational complexity of "hard" problems, often from number theory . For example, the hardness of RSA is related to the integer factorization problem, while Diffie–Hellman and DSA are related to the discrete logarithm problem. The security of elliptic curve cryptography is based on number theoretic problems involving elliptic curves . Because of the difficulty of the underlying problems, most public-key algorithms involve operations such as modular multiplication and exponentiation, which are much more computationally expensive than

5992-440: The discrete uniform distribution is a symmetric probability distribution wherein each of some finite whole number n of outcome values are equally likely to be observed. Thus every one of the n outcome values has equal probability 1/ n . Intuitively, a discrete uniform distribution is "a known, finite number of outcomes all equally likely to happen." A simple example of the discrete uniform distribution comes from throwing

6099-508: The one-time pad , are much more difficult to use in practice than the best theoretically breakable but computationally secure schemes. The growth of cryptographic technology has raised a number of legal issues in the Information Age . Cryptography's potential for use as a tool for espionage and sedition has led many governments to classify it as a weapon and to limit or even prohibit its use and export. In some jurisdictions where

6206-635: The rāz-saharīya which was used to communicate secret messages with other countries. David Kahn notes in The Codebreakers that modern cryptology originated among the Arabs , the first people to systematically document cryptanalytic methods. Al-Khalil (717–786) wrote the Book of Cryptographic Messages , which contains the first use of permutations and combinations to list all possible Arabic words with and without vowels. Ciphertexts produced by

6313-437: The security level of the underlying block cipher when the number of bits output from this PRNG is greater than two to the power of the underlying block cipher's block size in bits. When the maximum number of bits output from this PRNG is equal to the 2, the resulting output delivers the mathematically expected security level that the key size would be expected to generate, but the output is shown to not be indistinguishable from

6420-558: The 20th century, and several patented, among them rotor machines —famously including the Enigma machine used by the German government and military from the late 1920s and during World War II . The ciphers implemented by better quality examples of these machine designs brought about a substantial increase in cryptanalytic difficulty after WWI. Cryptanalysis of the new mechanical ciphering devices proved to be both difficult and laborious. In

6527-503: The Government Communications Headquarters ( GCHQ ), a British intelligence organization, revealed that cryptographers at GCHQ had anticipated several academic developments. Reportedly, around 1970, James H. Ellis had conceived the principles of asymmetric key cryptography. In 1973, Clifford Cocks invented a solution that was very similar in design rationale to RSA. In 1974, Malcolm J. Williamson

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6634-583: The Kautiliyam, the cipher letter substitutions are based on phonetic relations, such as vowels becoming consonants. In the Mulavediya, the cipher alphabet consists of pairing letters and using the reciprocal ones. In Sassanid Persia , there were two secret scripts, according to the Muslim author Ibn al-Nadim : the šāh-dabīrīya (literally "King's script") which was used for official correspondence, and

6741-402: The NIST announced that Keccak would be the new SHA-3 hash algorithm. Unlike block and stream ciphers that are invertible, cryptographic hash functions produce a hashed output that cannot be used to retrieve the original input data. Cryptographic hash functions are used to verify the authenticity of data retrieved from an untrusted source or to add a layer of security. The goal of cryptanalysis

6848-399: The NIST draft security standard approved for worldwide use in 2006. The leaked document states that "eventually, NSA became the sole editor". In spite of the known potential for a kleptographic backdoor and other known significant deficiencies with Dual_EC_DRBG, several companies such as RSA Security continued using Dual_EC_DRBG until the backdoor was confirmed in 2013. RSA Security received

6955-401: The NSA to readily decrypt material that was encrypted with the aid of Dual EC DRBG . Both papers reported that, as independent security experts long suspected, the NSA had been introducing weaknesses into CSPRNG standard 800-90; this being confirmed for the first time by one of the top-secret documents leaked to The Guardian by Edward Snowden . The NSA worked covertly to get its own version of

7062-660: The United Kingdom, cryptanalytic efforts at Bletchley Park during WWII spurred the development of more efficient means for carrying out repetitive tasks, such as military code breaking (decryption) . This culminated in the development of the Colossus , the world's first fully electronic, digital, programmable computer, which assisted in the decryption of ciphers generated by the German Army's Lorenz SZ40/42 machine. Extensive open academic research into cryptography

7169-494: The amount of randomness, the entropy that can be generated, is equal to the entropy provided by the system. But sometimes, in practical situations, numbers are needed with more randomness than the available entropy can provide. Also, the processes to extract randomness from a running system are slow in actual practice. In such instances, a CSPRNG can sometimes be used. A CSPRNG can "stretch" the available entropy over more bits. The requirements of an ordinary PRNG are also satisfied by

7276-474: The amusement of literate observers rather than as a way of concealing information. The Greeks of Classical times are said to have known of ciphers (e.g., the scytale transposition cipher claimed to have been used by the Spartan military). Steganography (i.e., hiding even the existence of a message so as to keep it confidential) was also first developed in ancient times. An early example, from Herodotus ,

7383-453: The combined study of cryptography and cryptanalysis. English is more flexible than several other languages in which "cryptology" (done by cryptologists) is always used in the second sense above. RFC   2828 advises that steganography is sometimes included in cryptology. The study of characteristics of languages that have some application in cryptography or cryptology (e.g. frequency data, letter combinations, universal patterns, etc.)

7490-428: The cryptanalytically uninformed. It was finally explicitly recognized in the 19th century that secrecy of a cipher's algorithm is not a sensible nor practical safeguard of message security; in fact, it was further realized that any adequate cryptographic scheme (including ciphers) should remain secure even if the adversary fully understands the cipher algorithm itself. Security of the key used should alone be sufficient for

7597-655: The earliest may have been the scytale of ancient Greece, a rod supposedly used by the Spartans as an aid for a transposition cipher. In medieval times, other aids were invented such as the cipher grille , which was also used for a kind of steganography. With the invention of polyalphabetic ciphers came more sophisticated aids such as Alberti's own cipher disk , Johannes Trithemius ' tabula recta scheme, and Thomas Jefferson 's wheel cypher (not publicly known, and reinvented independently by Bazeries around 1900). Many mechanical encryption/decryption devices were invented early in

7704-612: The early 20th century, cryptography was mainly concerned with linguistic and lexicographic patterns. Since then cryptography has broadened in scope, and now makes extensive use of mathematical subdisciplines, including information theory, computational complexity , statistics, combinatorics , abstract algebra , number theory , and finite mathematics . Cryptography is also a branch of engineering, but an unusual one since it deals with active, intelligent, and malevolent opposition; other kinds of engineering (e.g., civil or chemical engineering) need deal only with neutral natural forces. There

7811-409: The effort. Cryptographic hash functions are a third type of cryptographic algorithm. They take a message of any length as input, and output a short, fixed-length hash , which can be used in (for example) a digital signature. For good hash functions, an attacker cannot find two messages that produce the same hash. MD4 is a long-used hash function that is now broken; MD5 , a strengthened variant of MD4,

7918-549: The encryption and decryption algorithms that correspond to each key. Keys are important both formally and in actual practice, as ciphers without variable keys can be trivially broken with only the knowledge of the cipher used and are therefore useless (or even counter-productive) for most purposes. Historically, ciphers were often used directly for encryption or decryption without additional procedures such as authentication or integrity checks. There are two main types of cryptosystems: symmetric and asymmetric . In symmetric systems,

8025-506: The encryption and the reversing decryption. The detailed operation of a cipher is controlled both by the algorithm and, in each instance, by a "key". The key is a secret (ideally known only to the communicants), usually a string of characters (ideally short so it can be remembered by the user), which is needed to decrypt the ciphertext. In formal mathematical terms, a " cryptosystem " is the ordered list of elements of finite possible plaintexts, finite possible cyphertexts, finite possible keys, and

8132-487: The field since polyalphabetic substitution emerged in the Renaissance". In public-key cryptosystems, the public key may be freely distributed, while its paired private key must remain secret. In a public-key encryption system, the public key is used for encryption, while the private or secret key is used for decryption. While Diffie and Hellman could not find such a system, they showed that public-key cryptography

8239-715: The following sense. If the initial state s 1 {\displaystyle s_{1}} is chosen uniformly at random from { 0 , 1 } k {\displaystyle \{0,1\}^{k}} , then for any i , the sequence ( y 1 , y 2 , … , y i , s i + 1 ) {\displaystyle (y_{1},y_{2},\dots ,y_{i},s_{i+1})} must be computationally indistinguishable from ( r 1 , r 2 , … , r i , s i + 1 ) {\displaystyle (r_{1},r_{2},\dots ,r_{i},s_{i+1})} , in which

8346-406: The foundations of modern cryptography and provided a mathematical basis for future cryptography. His 1949 paper has been noted as having provided a "solid theoretical basis for cryptography and for cryptanalysis", and as having turned cryptography from an "art to a science". As a result of his contributions and work, he has been described as the "founding father of modern cryptography". Prior to

8453-415: The frequency analysis technique until the development of the polyalphabetic cipher , most clearly by Leon Battista Alberti around the year 1467, though there is some indication that it was already known to Al-Kindi. Alberti's innovation was to use different ciphers (i.e., substitution alphabets) for various parts of a message (perhaps for each successive plaintext letter at the limit). He also invented what

8560-548: The input string s i {\displaystyle s_{i}} with length k is the current state at period i , and the output ( s i + 1 {\displaystyle s_{i+1}} , y i {\displaystyle y_{i}} ) consists of the next state s i + 1 {\displaystyle s_{i+1}} and the pseudorandom output block y i {\displaystyle y_{i}} of period i , that withstands state compromise extensions in

8667-490: The key material comes from a true random source with high entropy, and thus any kind of pseudorandom number generator is insufficient. Ideally, the generation of random numbers in CSPRNGs uses entropy obtained from a high-quality source, generally the operating system's randomness API . However, unexpected correlations have been found in several such ostensibly independent processes. From an information-theoretic point of view,

8774-497: The key needed for decryption of that message). Encryption attempted to ensure secrecy in communications, such as those of spies , military leaders, and diplomats. In recent decades, the field has expanded beyond confidentiality concerns to include techniques for message integrity checking, sender/receiver identity authentication, digital signatures , interactive proofs and secure computation , among others. The main classical cipher types are transposition ciphers , which rearrange

8881-493: The meaning of encrypted information without access to the key normally required to do so; i.e., it is the study of how to "crack" encryption algorithms or their implementations. Some use the terms "cryptography" and "cryptology" interchangeably in English, while others (including US military practice generally) use "cryptography" to refer specifically to the use and practice of cryptographic techniques and "cryptology" to refer to

8988-462: The message. Most ciphers , apart from the one-time pad, can be broken with enough computational effort by brute force attack , but the amount of effort needed may be exponentially dependent on the key size, as compared to the effort needed to make use of the cipher. In such cases, effective security could be achieved if it is proven that the effort required (i.e., "work factor", in Shannon's terms)

9095-417: The modern age was effectively synonymous with encryption , converting readable information ( plaintext ) to unintelligible nonsense text ( ciphertext ), which can only be read by reversing the process ( decryption ). The sender of an encrypted (coded) message shares the decryption (decoding) technique only with the intended recipients to preclude access from adversaries. The cryptography literature often uses

9202-681: The names "Alice" (or "A") for the sender, "Bob" (or "B") for the intended recipient, and "Eve" (or "E") for the eavesdropping adversary. Since the development of rotor cipher machines in World War ;I and the advent of computers in World War II , cryptography methods have become increasingly complex and their applications more varied. Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions , making such algorithms hard to break in actual practice by any adversary. While it

9309-553: The notion of public-key (also, more generally, called asymmetric key ) cryptography in which two different but mathematically related keys are used—a public key and a private key. A public key system is so constructed that calculation of one key (the 'private key') is computationally infeasible from the other (the 'public key'), even though they are necessarily related. Instead, both keys are generated secretly, as an interrelated pair. The historian David Kahn described public-key cryptography as "the most revolutionary new concept in

9416-448: The older DES ( Data Encryption Standard ). Insecure symmetric algorithms include children's language tangling schemes such as Pig Latin or other cant , and all historical cryptographic schemes, however seriously intended, prior to the invention of the one-time pad early in the 20th century. In colloquial use, the term " code " is often used to mean any method of encryption or concealment of meaning. However, in cryptography, code has

9523-432: The only ones known until the 1970s, the same secret key encrypts and decrypts a message. Data manipulation in symmetric systems is significantly faster than in asymmetric systems. Asymmetric systems use a "public key" to encrypt a message and a related "private key" to decrypt it. The advantage of asymmetric systems is that the public key can be freely published, allowing parties to establish secure communication without having

9630-543: The order of letters in a message (e.g., 'hello world' becomes 'ehlol owrdl' in a trivially simple rearrangement scheme), and substitution ciphers , which systematically replace letters or groups of letters with other letters or groups of letters (e.g., 'fly at once' becomes 'gmz bu podf' by replacing each letter with the one following it in the Latin alphabet ). Simple versions of either have never offered much confidentiality from enterprising opponents. An early substitution cipher

9737-407: The output is not completely determined by their initial state. This addition aims to prevent attacks even if the initial state is compromised. Several CSPRNGs have been standardized. For example: The third PRNG in this standard, CTR_DRBG , is based on a block cipher running in counter mode . It has an uncontroversial design but has been proven to be weaker in terms of distinguishing attack, than

9844-449: The possible keys, to reach a point at which chances are better than even that the key sought will have been found. But this may not be enough assurance; a linear cryptanalysis attack against DES requires 2 known plaintexts (with their corresponding ciphertexts) and approximately 2 DES operations. This is a considerable improvement over brute force attacks. Uniform distribution (discrete) In probability theory and statistics ,

9951-654: The rest of the encryption parameters and deduce the master encryption key used to encrypt web sessions or virtual private network (VPN) connections." During World War II , Japan used a cipher machine for diplomatic communications; the United States was able to crack it and read its messages , mostly because the "key values" used were insufficiently random. Cryptography This is an accepted version of this page Cryptography , or cryptology (from Ancient Greek : κρυπτός , romanized :  kryptós "hidden, secret"; and γράφειν graphein , "to write", or -λογία -logia , "study", respectively ),

10058-551: The scope of brute-force attacks , so when specifying key lengths , the required key lengths are similarly advancing. The potential impact of quantum computing are already being considered by some cryptographic system designers developing post-quantum cryptography. The announced imminence of small implementations of these machines may be making the need for preemptive caution rather more than merely speculative. Claude Shannon 's two papers, his 1948 paper on information theory , and especially his 1949 paper on cryptography, laid

10165-410: The sender and receiver share the same key (or, less commonly, in which their keys are different, but related in an easily computable way). This was the only kind of encryption publicly known until June 1976. Symmetric key ciphers are implemented as either block ciphers or stream ciphers . A block cipher enciphers input in blocks of plaintext as opposed to individual characters, the input form used by

10272-399: The six-sided die could have abstract symbols rather than numbers on each of its faces. Less simply, a random permutation is a permutation generated uniformly randomly from the permutations of a given set and a uniform spanning tree of a graph is a spanning tree selected with uniform probabilities from the full set of spanning trees of the graph. The discrete uniform distribution itself

10379-408: The technique is easily generalized to any block cipher; AES has been suggested. If the key k is leaked, the entire X9.17 stream can be predicted; this weakness is cited as a reason for creating Yarrow. All these above-mentioned schemes, save for X9.17, also mix the state of a CSPRNG with an additional source of entropy. They are therefore not "pure" pseudorandom number generators, in the sense that

10486-412: The techniques used in most block ciphers, especially with typical key sizes. As a result, public-key cryptosystems are commonly hybrid cryptosystems , in which a fast high-quality symmetric-key encryption algorithm is used for the message itself, while the relevant symmetric key is sent with the message, but encrypted using a public-key algorithm. Similarly, hybrid signature schemes are often used, in which

10593-508: The traffic and then forward it to the recipient. Also important, often overwhelmingly so, are mistakes (generally in the design or use of one of the protocols involved). Cryptanalysis of symmetric-key ciphers typically involves looking for attacks against the block ciphers or stream ciphers that are more efficient than any attack that could be against a perfect cipher. For example, a simple brute force attack against DES requires one known plaintext and 2 decryptions, trying approximately half of

10700-431: The use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation. Cryptography also plays a major role in digital rights management and copyright infringement disputes with regard to digital media . The first use of the term "cryptograph" (as opposed to " cryptogram ") dates back to the 19th century—originating from " The Gold-Bug ",

10807-525: Was a message tattooed on a slave's shaved head and concealed under the regrown hair. Other steganography methods involve 'hiding in plain sight,' such as using a music cipher to disguise an encrypted message within a regular piece of sheet music. More modern examples of steganography include the use of invisible ink , microdots , and digital watermarks to conceal information. In India, the 2000-year-old Kama Sutra of Vātsyāyana speaks of two different kinds of ciphers called Kautiliyam and Mulavediya. In

10914-542: Was finally won in 1978 by Ronald Rivest , Adi Shamir , and Len Adleman , whose solution has since become known as the RSA algorithm . The Diffie–Hellman and RSA algorithms , in addition to being the first publicly known examples of high-quality public-key algorithms, have been among the most widely used. Other asymmetric-key algorithms include the Cramer–Shoup cryptosystem , ElGamal encryption , and various elliptic curve techniques . A document published in 1997 by

11021-499: Was first published about ten years later by Friedrich Kasiski . Although frequency analysis can be a powerful and general technique against many ciphers, encryption has still often been effective in practice, as many a would-be cryptanalyst was unaware of the technique. Breaking a message without using frequency analysis essentially required knowledge of the cipher used and perhaps of the key involved, thus making espionage, bribery, burglary, defection, etc., more attractive approaches to

11128-488: Was indeed possible by presenting the Diffie–Hellman key exchange protocol, a solution that is now widely used in secure communications to allow two parties to secretly agree on a shared encryption key . The X.509 standard defines the most commonly used format for public key certificates . Diffie and Hellman's publication sparked widespread academic efforts in finding a practical public-key encryption system. This race

11235-570: Was new and significant. Computer use has thus supplanted linguistic cryptography, both for cipher design and cryptanalysis. Many computer ciphers can be characterized by their operation on binary bit sequences (sometimes in groups or blocks), unlike classical and mechanical schemes, which generally manipulate traditional characters (i.e., letters and digits) directly. However, computers have also assisted cryptanalysis, which has compensated to some extent for increased cipher complexity. Nonetheless, good modern ciphers have stayed ahead of cryptanalysis; it

11342-518: Was probably the first automatic cipher device , a wheel that implemented a partial realization of his invention. In the Vigenère cipher , a polyalphabetic cipher , encryption uses a key word , which controls letter substitution depending on which letter of the key word is used. In the mid-19th century Charles Babbage showed that the Vigenère cipher was vulnerable to Kasiski examination , but this

11449-533: Was the Caesar cipher , in which each letter in the plaintext was replaced by a letter three positions further down the alphabet. Suetonius reports that Julius Caesar used it with a shift of three to communicate with his generals. Atbash is an example of an early Hebrew cipher. The earliest known use of cryptography is some carved ciphertext on stone in Egypt ( c.  1900 BCE ), but this may have been done for

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