Accepted in Advances in Mathematics of Communications.

## (Not So) Random Shuffles of RC4

Journal of Mathematical Cryptology. Volume 11, Number 2, pp. A Differential Fault Attack on Plantlet. Volume 66, Number 10, pp. Designs, Codes and Cryptography. Volume 82, Number , pp. Cryptography and Communications. Volume 9, Number 4, pp. Discrete Mathematics. Volume , Number 3, pp. Discrete Applied Mathematics.

Volume , pp. Volume , Number 1, pp. Volume 7, Number 3, pp. Volume 73, Number 2, pp. Differential Fault Attack against Grain family with very few faults and minimal assumptions. Dobb's Journal. The book details how programmers and electronic communications professionals can use cryptography-the technique of enciphering and deciphering messages-to maintain the privacy of computer data.

It describes dozens of cryptography algorithms, gives practical advice on how to implement them into cryptographic software, and shows how they can be used to solve security problems. The book shows programmers who design computer applications, networks, and storage systems how they can build security into their software and systems. With a new Introduction by the author, this premium edition will be a keepsake for all those committed to computer and cyber security.

Stay ahead with the world's most comprehensive technology and business learning platform. 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. Ellis had conceived the principles of asymmetric key cryptography. Williamson 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.

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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 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. 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. 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 the techniques used in most block ciphers, especially with typical key sizes.

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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.

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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.

## An Overview of Cryptography

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 with their corresponding ciphertexts and approximately 2 43 DES operations. Public-key algorithms are based on the computational difficulty of various problems. The most famous of these are the difficulty of integer factorization of semiprimes and the difficulty of calculating discrete logarithms , both of which are not yet proven to be solvable in polynomial time using only a classical Turing-complete computer.

Much public-key cryptanalysis concerns designing algorithms in P that can solve these problems, or using other technologies, such as quantum computers.

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.

## RC4 stream cipher and its variants.

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 [52] 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 humans e.

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

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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 e. 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 [53] systems, signcryption systems, etc. Some more 'theoretical' [ clarification needed ] cryptosystems include interactive proof systems , [54] like zero-knowledge proofs , [55] systems for secret sharing , [56] [57] etc. 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.