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relevant challenges. With the help of cryptography, many of these challenges can be . for scientific methods in cryptography, a strong connection to mathematics and a To base definitions of security on games was a landmark idea.  Neal Koblitz, The uneasy relationship between mathematics and cryptography. The Uneasy Relationship Between. Mathematics and Cryptography . women with energy and drive a chance to get back in the game. The "uneasy" relationship between mathematics and cryptography or a football game mean you have to risk personal harms like loss of.
A generic fifth participant, but rarely used, as "E" is usually reserved for Eve. An eavesdropperwho is usually a passive attacker.
While she can listen in on messages between Alice and Bob, she cannot modify them. In quantum cryptographyEve may also represent the environment[ clarification needed ]. A trusted advisorcourier or intermediary. Faythe is used infrequently, and is associated with Faith and Faithfulness. Faythe may be a repository of key service or courier of shared secrets. A generic sixth participant.
For example, Grace may try to force Alice or Bob to implement backdoors in their protocols. May also deliberately weaken standards.
A mischievous designer for cryptographic standards, but rarely used. A judge who may be called upon to resolve a potential dispute between participants. Mallory    or less commonly Mallet     A malicious attacker.
Associated with Trudy, an intruder. Used as an alternative to the eavesdropper Eve. Used as an alternative to the eavesdropper Eve in several South Asian nations Olivia. An oracle, who provides external data to smart contracts residing on distributed ledger commonly referred to as blockchain systems. An opponent, similar to Mallory, but not necessarily malicious. A prover, who interacts with the system to show that the intended transaction has actually taken place.
Peggy is often found in zero-knowledge proofs. Similar to Victor or Vanna.
The uneasy relationship – Let Them Dance
An pseudonymous attacker, who usually uses a large number of identities. For example, Sybil may attempt to subvert a reputation system.
A trusted arbitratorwho acts as a neutral third party. Victor,  or Vanna. A wardenwho may guard Alice and Bob. A whistleblowerwho is an insider with privileged access capable of divulging information. For interactive proof systems there are other characters: Then I must keep finding ways to stay a step ahead of my "enemy. However, usually a cipher refers to replacing a symbol of a plaintext alphabet by another single symbol from some other alphabet or the same alphabet.
By contrast, a code refers to replacing blocks of symbols in the plaintext, by another block of symbols. There have been many developments in cryptography since the Caesar Cipher. One simple idea is not to use the same word lengths in the enciphered message as in the original.
If the word "I" or "a" is left as a single letter in the "ciphertext," then it greatly simplifies the process of breaking the encryption system.
Typically, the message is broken up into groups of 5 letters disregarding spaces between words, and often punctuation and replaced by 5 other letters. This has the effect of making the coded message look more "anonymous. Another simple idea is the use of a polyalphabetic cipherwhere the alphabet used to encode the plaintext changes with each letter in accordance with some key. Using the key provides a way to change the alphabet used for the encoding as one matches a plaintext letter to a letter in the key.
This idea was pioneered by Leone Albertiwho was also a pioneer of projective geometry. It is tempting to believe that such a "complex" system would be unbreakable. However, if the key length is short and there is lots of ciphertext available using the same system, then statistical methods can be used to break the cipher. If a key is used only once and is generated at random, the so-called one-time pad, then the cipher is not breakable. However, key exchange and generation of large amounts of random key present a significant problem for the volume of communications that we want to secure in modern times.
Skipping to more modern times we come to a very important period and figure for the development of mathematics' role in security issues, Alan Turing After Hitler's forces took over countries such as Holland and France, Germany wanted to extend its conquests to include Britain. Britain began an elaborate effort to try to prevent invasion of the British Isles by taking advantage of communications and signals intelligence patterns in levels of communication traffic that might indicate some special military operation and information obtained by deciphering coded communication.
A special group of individuals was assembled at Bletchley Parkexperts in languages and mathematics, to try to glean as much useable information from the enemy as possible.
The first successes in breaking German codes was due to the work of Polish mathematiciansincluding work of Marian Rejewskiwho were able to hand over what they had achieved to the British.
The Uneasy Relationship Between Mathematics and Cryptography [PDF] : compsci
Photo courtesy of NSA, the caption reads "Marian Rejewski, the Polish mathematician who made the initial breakthrough against the Enigma machine. Photo courtesy of NSA Turing worked with the mathematician Gordon Welchman to develop a specialized "computer" to help break the Enigma-generated codes.
Bombe courtesy of NSA In the United States too, many men and womenincluding many linguists and mathematicians, were involved with the war effort. Among the most famous of these individuals were William Friedman who used mathematical techniques though not trained as a mathematician and his wife Elizabeth a linguist.
The United States' team broke many Japanese codes and thereby were able to change the course of the war. Elizabeth Friedman and William Friedman: Photos courtesy of NSA Modern cryptography is no longer primarily the concern of the military and diplomats. Cryptography is increasingly applied to maintain capitalism's infrastructure: When money is transferred electronically, when an email is sent, when a purchase is made online, the users of such systems want to know that the transactions went as planned and were not "hijacked.
When one party sends a message to another party, one would like to think: The world of "intrigue" that one associates with spies and spying plays out in with no less complexity in the world of Internet security. Hashing Hashing is the process of taking a long string and replacing it by a much shorter string in some systematic way.
For example, if one started with a poem, one could replace it by the string which gives the number of letters in the poem. At first glance it might appear that hashing involves issues of data compression rather than data security. However, it should also be clear that a hashed string might have cryptological importance because it might be possible to build into the hashing system that it be hard to figure out what the original string was.
The complication with hashing is that ideally one wants to avoid having different strings hash to the same string. When this happens one says that a collision has occurred. In the system described above, two poems of exactly the same number of letters would create a collision.
Thus, one has to build into one's system what to do about collisions, should they occur. Another desirable feature of hashing designed for security purposes is that when two similar strings are hashed, the results are extremely different. Thus, if someone tries to modify a "secure document" in some small way, that the "intrusion" could be detected because the hash of the two documents, the original and the forged one, would be very different.
There is a circle of ideas that's involved in why hashing is very closely tied with the issue of passwords and digital signatures. A digital signature is an electronic identification system analogous to the handwritten signatures commonly used for letters and checks.
One wants to have a system which minimizes the dangers of forgery. However, in in a series of developments let by mathematicians and computer scientists, it became clear that MD-5 had problems with its security, so it was commonly replaced by SHA Hashing has had an intimate connection with Internet security in that it is a commonly used feature of password management systems. Passwords are not only used for email accounts but they are often used to access commercial services that an individual might subscribe to using the Internet e.
One favor you can do for yourself is to practice some simple rules for improving password security over the ones that many individuals use. Several standards for hashing systems have unfortunately, in light of concerns about maintaining security, proved to be breakable with available computing resource systems.
Recently, SHA-1, considered the workhorse of secure hashing systems, was shown to be vulnerable. This work was done by a Chinese mathematician, Xiaoyun Wangand her co-workers. There is disagreement about the short term implications of Wang's work. Though it does not appear that available computer resources make it possible to break SHA-1, the progress made by Wang and her colleagues raises the issue that some clever variant of the ideas she has pioneered will place existing systems in danger.
More secure systems than SHA-1 are currently known but they are not as speedy. It might seem that there are so many ways that a crypto-system could be implemented that it would be hopeless to discover the system that was being used and be able to recover the plaintext message from the coded message. There are two common ways to break crypto-systems. One is to use statistical techniques. If one has large collections of "ciphertext," one can look for patterns using frequency distributions for the symbols that appear to try to break the system.
The other method for breaking the system is that people who use crypto-systems often have "tics" or "mannerisms" that give entry points into deciphering the systems. Perhaps some particular user always begins the message with the same greeting. Perhaps a message always begins with a weather report which allows one to try to guess the way common words are encoded. Another factor is that when a system is widely used, there are typically many people involved in its management.
In such cases it is difficult to prevent information about the type of system being used from becoming known.
Thus, the methods that are used for attacking substitution ciphers are different from the methods that are based on matrix techniques. Given the practical realities involved, it is widely felt that the proper way to view security is to assume that the person trying to break one's security system knows exactly the type of system one is using. Thus, the "enemy" might know that you are using RSA explained below or that you are using a Hill Cipher matrix oriented system.
Public Key Systems A major revolution in cryptology occurred in the middle 's with the discovery some say rediscovery of a new paradigm for codes.
In traditional cryptography the two parties that wanted to share secret information arranged a system, which typically required a "key exchange," of a single key. Intuitively, think of a key as a way to "lock" a message that is being sent in secret.
If the receiver has an identical key to the sender, then the message can be unlocked. Thus, single key systems involve having a common basis key for the person who encrypted a message and the person who decrypted it to operate the crypto-system. All this changed with the development of Public Key Cryptography. Credit for this development is usually given to Ralph MerkleWhitfield Diffie and Martin Hellmanthough the history of this development is complex.
Whitfield Diffie Courtesy of Dr. Diffie and Sun MicrosystemsProf. Martin Hellman Courtesy of Dr. Hellmanand Dr. Ralph Merkle Courtesy of Dr. Merkle Public Key Cryptology is based on having two keys.
One key, used to send a secret message to a particular person X, is publicly available, like a phone number in telephone directory. The second key, which is not made public, is held by X to be used in conjunction with the public key. Another aspect of public key cryptography and modern private key systems is having a method of allowing strangers to confidently exchange keys with each other. Such a system was devised by Diffie and Hellman using ideas from Merkle and it is designed to work over a nonsecure communication system.
This system was patentedthough the patent has now expired. Below we will discuss in a bit more detail the best known of the public key systems, known as as RSA. Another popular public key system is due to Taher Elgamal This method uses ideas about group theory and complexity issues for its security. Photo courtesy of Dr. Two integers a and b are said to be congruent modulo m a positive integer, which is at least 2 written: Here are some examples: Note that we can always arrange the number on the right hand side of a congruence to be a number between 0 and m-1 where m is the modulus.
Thus, we could replace 23 by 10 in the last congruence. It is not very difficult to find the value for the "? The idea for doing this is to compute the values of 5, 52, 54, 58, etc. However, the problem of finding the value of k for which the congruence below is valid is much less straightforward: The problem of finding k in a situation such as this is known as the discrete logarithm problem. When the modulus is very large, methods which are appreciably better than brute force are not currently known.
The complexity of finding discrete logarithms for various m, in particular, when m is prime and many other algorithms that have been used to try to design public key systems, is not fully understood. It turns out that some systems based on NP-complete problems have been " broken " while other systems which depend on problems whose complexity is still not understood fully seem to be holding their own.