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This P is a large prime number of over digits. Let us now assume we have two other integer s, a and b. Now say we want to find the value of N, so that value is found by the following formula:.

This is known as discrete exponentiation and is quite simple to compute. However, the opposite is true when we invert it.

If we are given P, a, and N and are required to find b so that the equation is valid, then we face a tremendous level of difficulty.

This problem forms the basis for a number of public key infrastructure algorithms, such as Diffie-Hellman and EIGamal.

This problem has been studied for many years and cryptography based on it has withstood many forms of attacks.

The Integer Factorization Problem: This is simple in concept. Say that one takes two prime numbers, P2 and P1, which are both "large" a relative term, the definition of which continues to move forward as computing power increases.

We then multiply these two primes to produce the product, N. The difficulty arises when, being given N, we try and find the original P1 and P2. The Rivest-Shamir-Adleman public key infrastructure encryption protocol is one of many based on this problem.

To simplify matters to a great degree, the N product is the public key and the P1 and P2 numbers are, together, the private key. This problem is one of the most fundamental of all mathematical concepts.

It has been studied intensely for the past 20 years and the consensus seems to be that there is some unproven or undiscovered law of mathematics that forbids any shortcuts.

That said, the mere fact that it is being studied intensely leads many others to worry that, somehow, a breakthrough may be discovered.

This is a new cryptographic protocol based upon a reasonably well-known mathematical problem. The properties of elliptic curves have been well known for centuries, but it is only recently that their application to the field of cryptography has been undertaken.

First, imagine a huge piece of paper on which is printed a series of vertical and horizontal lines. Each line represents an integer with the vertical lines forming x class components and horizontal lines forming the y class components.

The intersection of a horizontal and vertical line gives a set of coordinates x,y. In the highly simplified example below, we have an elliptic curve that is defined by the equation:.

For the above, given a definable operator, we can determine any third point on the curve given any two other points. This definable operator forms a "group" of finite length.

To add two points on an elliptic curve, we first need to understand that any straight line that passes through this curve intersects it at precisely three points.

Now, say we define two of these points as u and v: We can then draw a vertical line through w to find the final intersecting point at x.

This rule works, when we define another imaginary point, the Origin, or O, which exists at theoretically extreme points on the curve.

As strange as this problem may seem, it does permit for an effective encryption system, but it does have its detractors.

On the positive side, the problem appears to be quite intractable, requiring a shorter key length thus allowing for quicker processing time for equivalent security levels as compared to the Integer Factorization Problem and the Discrete Logarithm Problem.

On the negative side, critics contend that this problem, since it has only recently begun to be implemented in cryptography, has not had the intense scrutiny of many years that is required to give it a sufficient level of trust as being secure.

This leads us to more general problem of cryptology than of the intractability of the various mathematical concepts, which is that the more time, effort, and resources that can be devoted to studying a problem, then the greater the possibility that a solution, or at least a weakness, will be found.

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Learn how his team is prepping for the According to recent industry analyst predictions, digital transformation budgets are on the rise for , with investments in One of his tips: Think data quality, not The software that manages the IT help desk must allow admins to track, monitor and resolve incidents to optimize the benefits IT professionals should look to third-party migration tools during a transition to Windows 10 to provide any features that the A Windows reboot loop is a vicious and frustrating cycle, but there are ways you can fix the problem, including booting in Safe 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 is typically the case that use of a quality cipher is very efficient i.

Extensive open academic research into cryptography is relatively recent; it began only in the mids. In recent times, IBM personnel designed the algorithm that became the Federal i.

Following their work in , it became popular to consider cryptography systems based on mathematical problems that are easy to state but have been found difficult to solve.

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 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. 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 the scope of brute-force attacks , so when specifying key lengths , the required key lengths are similarly advancing.

Essentially, prior to the early 20th century, cryptography was chiefly concerned with linguistic and lexicographic patterns. Since then the emphasis has shifted, and cryptography now makes extensive use of mathematics, including aspects of information theory , computational complexity , statistics , combinatorics , abstract algebra , number theory , and finite mathematics generally.

Cryptography is also a branch of engineering , but an unusual one since it deals with active, intelligent, and malevolent opposition see cryptographic engineering and security engineering ; other kinds of engineering e.

There is also active research examining the relationship between cryptographic problems and quantum physics see quantum cryptography and quantum computer.

The modern field of cryptography can be divided into several areas of study. The chief ones are discussed here; see Topics in Cryptography for more.

Symmetric-key cryptography refers to encryption methods in which both 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 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 a stream cipher. 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 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; see Category: 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, is also widely used but broken in practice.

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 it isn't yet widely deployed; 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.

Cryptographic hash functions are used to verify the authenticity of data retrieved from an untrusted source or to add a layer of security.

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; [4] this additional complication blocks an attack scheme against bare digest algorithms , and so has been thought worth the effort.

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 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.

The difficulty of securely establishing a secret key between two communicating parties, when a secure channel does not already exist between them, also presents a chicken-and-egg problem which is a considerable practical obstacle for cryptography users in the real world.

In a groundbreaking paper, Whitfield Diffie and Martin Hellman proposed 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.

Instead, both keys are generated secretly, as an interrelated pair. 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 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.

Diffie and Hellman's publication sparked widespread academic efforts in finding a practical public-key encryption system.

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. To much surprise, a document published in by the Government Communications Headquarters GCHQ , a British intelligence organization, revealed that cryptographers at GCHQ had anticipated several academic developments.

Ellis had conceived the principles of asymmetric key cryptography. Williamson is claimed to have developed the Diffie—Hellman key exchange.

Public-key cryptography can also be 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: Digital signatures are central to the operation of public key infrastructures and many network security schemes e.

Public-key algorithms are most often based on 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.

More recently, elliptic curve cryptography has developed, a system in which security 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 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 a cryptographic hash function is computed, and only the resulting hash is digitally signed.

The goal of cryptanalysis 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 the message.

In such cases, effective security could be achieved if it is proven that the effort required i. 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. There are 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 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. 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 55 decryptions, trying approximately half of 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 43 known plaintexts and approximately 2 43 DES operations.

Public-key algorithms are based on the computational difficulty of various problems. The most famous of these is integer factorization e.

Much public-key cryptanalysis concerns numerical algorithms for solving these computational problems, or some of them, efficiently i.

For instance, the best known algorithms for solving the elliptic curve-based version of discrete logarithm are much more time-consuming than the best known algorithms for factoring, at least for problems of more or less equivalent size.

Thus, other things being equal, to achieve an equivalent strength of attack resistance, factoring-based encryption techniques must use larger keys than elliptic curve techniques.

For this reason, public-key cryptosystems based on elliptic curves have become popular since their invention in the mids.

While pure cryptanalysis uses weaknesses in the algorithms themselves, other attacks on cryptosystems are based on actual use of the algorithms in real devices, and are called side-channel attacks.

If a cryptanalyst has access to, for example, the amount of time the device took to encrypt a number of plaintexts or report an error in a password or PIN character, he may be able to use a timing attack to break a cipher that is otherwise resistant to analysis.

An attacker might also study the pattern and length of messages to derive valuable information; this is known as traffic analysis [45] and can be quite useful to an alert adversary.

Poor administration of a cryptosystem, such as permitting too short keys, will make any system vulnerable, regardless of other virtues.

Social engineering and other attacks against the personnel who work with cryptosystems or the messages they handle e.

Much of the theoretical work in cryptography concerns cryptographic primitives —algorithms with basic cryptographic properties—and their relationship to other cryptographic problems.

More complicated cryptographic tools are then built from these basic primitives. These primitives provide fundamental properties, which are used to develop more complex tools called cryptosystems or cryptographic protocols , which guarantee one or more high-level security properties.

Note however, that the distinction between cryptographic primitives and cryptosystems, is quite arbitrary; for example, the RSA algorithm is sometimes considered a cryptosystem, and sometimes a primitive.

Typical examples of cryptographic primitives include pseudorandom functions , one-way functions , etc. One or more cryptographic primitives are often used to develop a more complex algorithm, called a cryptographic system, or cryptosystem.

Cryptosystems use the properties of the underlying cryptographic primitives to support the system's security properties. As the distinction between primitives and cryptosystems is somewhat arbitrary, a sophisticated cryptosystem can be derived from a combination of several more primitive cryptosystems.

In many cases, the cryptosystem's structure involves back and forth communication among two or more parties in space e. Such cryptosystems are sometimes called cryptographic protocols.

More complex cryptosystems include electronic cash [46] systems, signcryption systems, etc. Some more 'theoretical' cryptosystems include interactive proof systems , [47] like zero-knowledge proofs , [48] systems for secret sharing , [49] [50] etc.

Until recently [ timeframe? The general idea of provable security is to give arguments about the computational difficulty needed to compromise some security aspect of the cryptosystem i.

The study of how best to implement and integrate cryptography in software applications is itself a distinct field see Cryptographic engineering and Security engineering.

Cryptography has long been of interest to intelligence gathering and law enforcement agencies. Because of its facilitation of privacy , and the diminution of privacy attendant on its prohibition, cryptography is also of considerable interest to civil rights supporters.

Accordingly, there has been a history of controversial legal issues surrounding cryptography, especially since the advent of inexpensive computers has made widespread access to high quality cryptography possible.

In some countries, even the domestic use of cryptography is, or has been, restricted. Until , France significantly restricted the use of cryptography domestically, though it has since relaxed many of these rules.

In China and Iran , a license is still required to use cryptography. In the United States , cryptography is legal for domestic use, but there has been much conflict over legal issues related to cryptography.

Probably because of the importance of cryptanalysis in World War II and an expectation that cryptography would continue to be important for national security, many Western governments have, at some point, strictly regulated export of cryptography.

After World War II, it was illegal in the US to sell or distribute encryption technology overseas; in fact, encryption was designated as auxiliary military equipment and put on the United States Munitions List.

However, as the Internet grew and computers became more widely available, high-quality encryption techniques became well known around the globe.

In the s, there were several challenges to US export regulation of cryptography. Bernstein , then a graduate student at UC Berkeley , brought a lawsuit against the US government challenging some aspects of the restrictions based on free speech grounds.

The case Bernstein v. United States ultimately resulted in a decision that printed source code for cryptographic algorithms and systems was protected as free speech by the United States Constitution.

In , thirty-nine countries signed the Wassenaar Arrangement , an arms control treaty that deals with the export of arms and "dual-use" technologies such as cryptography.

The treaty stipulated that the use of cryptography with short key-lengths bit for symmetric encryption, bit for RSA would no longer be export-controlled.

Since this relaxation in US export restrictions, and because most personal computers connected to the Internet include US-sourced web browsers such as Firefox or Internet Explorer , almost every Internet user worldwide has potential access to quality cryptography via their browsers e.

Many Internet users don't realize that their basic application software contains such extensive cryptosystems.

These browsers and email programs are so ubiquitous that even governments whose intent is to regulate civilian use of cryptography generally don't find it practical to do much to control distribution or use of cryptography of this quality, so even when such laws are in force, actual enforcement is often effectively impossible.

Another contentious issue connected to cryptography in the United States is the influence of the National Security Agency on cipher development and policy.

The technique became publicly known only when Biham and Shamir re-discovered and announced it some years later. The entire affair illustrates the difficulty of determining what resources and knowledge an attacker might actually have.

Another instance of the NSA's involvement was the Clipper chip affair, an encryption microchip intended to be part of the Capstone cryptography-control initiative.

Clipper was widely criticized by cryptographers for two reasons. The cipher algorithm called Skipjack was then classified declassified in , long after the Clipper initiative lapsed.

The classified cipher caused concerns that the NSA had deliberately made the cipher weak in order to assist its intelligence efforts. The whole initiative was also criticized based on its violation of Kerckhoffs's Principle , as the scheme included a special escrow key held by the government for use by law enforcement, for example in wiretaps.

Cryptography is central to digital rights management DRM , a group of techniques for technologically controlling use of copyrighted material, being widely implemented and deployed at the behest of some copyright holders.

President Bill Clinton signed the Digital Millennium Copyright Act DMCA , which criminalized all production, dissemination, and use of certain cryptanalytic techniques and technology now known or later discovered ; specifically, those that could be used to circumvent DRM technological schemes.

Similar statutes have since been enacted in several countries and regions, including the implementation in the EU Copyright Directive.

Similar restrictions are called for by treaties signed by World Intellectual Property Organization member-states. Niels Ferguson , a well-respected cryptography researcher, has publicly stated that he will not release some of his research into an Intel security design for fear of prosecution under the DMCA.

Dmitry Sklyarov was arrested during a visit to the US from Russia, and jailed for five months pending trial for alleged violations of the DMCA arising from work he had done in Russia, where the work was legal.

In both cases, the MPAA sent out numerous DMCA takedown notices, and there was a massive Internet backlash [9] triggered by the perceived impact of such notices on fair use and free speech.

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