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Cyclometer (horse)
The cyclometer was a cryptologic device designed, "probably in 1934 or 1935," by Marian Rejewski of the Polish Cipher Bureau's German section (BS-4) to facilitate decryption of German Enigma ciphertext. The original machines are believed to have been destroyed shortly before the German invasion of Poland that launched the Second World War, to prevent the Germans learning that their cipher had been broken. Together with Rejewski's later cryptologic bomb, it can be viewed as a predecessor to the Bombe that was later to break the Enigma code later in the war at Bletchley Park in England. Using drawings made by Rejewski, Hal Evans and Tim Flack at the Department of Engineering, University of Cambridge, in 2019 constructed a working version of the cyclometer. History Example message Fede Weierud provides the procedure, secret settings, and results that where use in a 1950 German technical manual. Daily key (shared secret): Wheel Order : II I III Ringstellung : 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Squaring The Circle
Squaring the circle is a problem in geometry first proposed in Greek mathematics. It is the challenge of constructing a square with the area of a circle by using only a finite number of steps with a compass and straightedge. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (\pi) is a transcendental number. That is, \pi is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if \pi were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. Despite the proof that it is impossible, attempts to square the circle have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henryk Zygalski
Henryk Zygalski (; 15 July 1908 – 30 August 1978) was a Polish mathematician and cryptologist who worked at breaking German Enigma ciphers before and during World War II. Life Zygalski was born on 15 July 1908 in Posen, German Empire (now Poznań, Poland). He was, from September 1932, a civilian cryptologist with the Polish General Staff's ''Biuro Szyfrów'' (Cipher Bureau), housed in the Saxon Palace in Warsaw. He worked there with fellow Poznań University alumni and Cipher Bureau cryptology-course graduates Marian Rejewski and Jerzy Różycki. Together they developed methods and equipment for breaking Enigma messages. In late 1938, in response to growing complexities in German encryption procedures, Zygalski designed the " perforated sheets," also known as " Zygalski sheets," a manual device for finding Enigma settings. This scheme, like the earlier "card catalog," was independent of the number of connections being used in the Enigma's plugboard, or commutator. After t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflector (cipher Machine)
A reflector, in cryptology, is a component of some rotor cipher machines, such as the Enigma machine, that sends electrical impulses that have reached it from the machine's rotors, back in reverse order through those rotors. The reflector simplified using the same machine setup for encryption and decryption, but it creates a weakness in the encryption: with a reflector the encrypted version of a given letter can never be that letter itself. That limitation aided World War II code breakers in cracking Enigma encryption. The comparable WW II U.S. cipher machine, SIGABA In the history of cryptography, the ECM Mark II was a cipher machine used by the United States for message encryption from World War II until the 1950s. The machine was also known as the SIGABA or Converter M-134 by the Army, or CSP-888/889 by the ..., did not include a reflector. Other names The reflector is also known as the reversing drum or, from the German, the ''Umkehrwalze'' or ''UKW''. {{crypto-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Card Catalog (cryptology)
The card catalog, or "catalog of characteristics," in cryptography, was a system designed by Polish Cipher Bureau mathematician-cryptologist Marian Rejewski, and first completed about 1935 or 1936, to facilitate decrypting German Enigma ciphers. History The Polish Cipher Bureau used the theory of permutations to start breaking the Enigma cipher in late 1932. The Bureau recognized that the Enigma machine's doubled-key (see Grill (cryptology)) permutations formed cycles, and those cycles could be used to break the cipher. With German cipher keys provided by a French spy, the Bureau was able to reverse engineer the Enigma and start reading German messages. At the time, the Germans were using only 6 ''steckers'', and the Polish grill method was feasible. On 1 August 1936, the Germans started using 8 ''steckers'', and that change made the grill method less feasible. The Bureau needed an improved method to break the German cipher. Although the ''steckers'' would change which lett ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cycle Graph (algebra)
In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups. A cycle is the set of powers of a given group element ''a'', where ''an'', the ''n''-th power of an element ''a'' is defined as the product of ''a'' multiplied by itself ''n'' times. The element ''a'' is said to ''generate'' the cycle. In a finite group, some non-zero power of ''a'' must be the group identity, ''e''; the lowest such power is the order of the cycle, the number of distinct elements in it. In a cycle graph, the cycle is represented as a polygon, with the vertices representing the group elements, and the connecting lines indicating that all elements in that polygon are members of the same cycle. Cycles Cycles can overlap, or they can have no element in common but the identity. The cycle graph displays each interesting cycle as a polygon. If ''a'' generates a cycle of order ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hash Table
In computing, a hash table, also known as hash map, is a data structure that implements an associative array or dictionary. It is an abstract data type that maps keys to values. A hash table uses a hash function to compute an ''index'', also called a ''hash code'', into an array of ''buckets'' or ''slots'', from which the desired value can be found. During lookup, the key is hashed and the resulting hash indicates where the corresponding value is stored. Ideally, the hash function will assign each key to a unique bucket, but most hash table designs employ an imperfect hash function, which might cause hash '' collisions'' where the hash function generates the same index for more than one key. Such collisions are typically accommodated in some way. In a well-dimensioned hash table, the average time complexity for each lookup is independent of the number of elements stored in the table. Many hash table designs also allow arbitrary insertions and deletions of key–value pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hash Function
A hash function is any function that can be used to map data of arbitrary size to fixed-size values. The values returned by a hash function are called ''hash values'', ''hash codes'', ''digests'', or simply ''hashes''. The values are usually used to index a fixed-size table called a '' hash table''. Use of a hash function to index a hash table is called ''hashing'' or ''scatter storage addressing''. Hash functions and their associated hash tables are used in data storage and retrieval applications to access data in a small and nearly constant time per retrieval. They require an amount of storage space only fractionally greater than the total space required for the data or records themselves. Hashing is a computationally and storage space-efficient form of data access that avoids the non-constant access time of ordered and unordered lists and structured trees, and the often exponential storage requirements of direct access of state spaces of large or variable-length keys. Use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partition (number Theory)
In number theory and combinatorics, a partition of a positive integer , also called an integer partition, is a way of writing as a summation, sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition (combinatorics), composition.) For example, can be partitioned in five distinct ways: : : : : : The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . A summand in a partition is also called a part. The number of partitions of is given by the partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials and of the symmetric group and in group representation, group representation theory in genera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grill (cryptology)
The grill method ( pl, metoda rusztu), in cryptology, was a method used chiefly early on, before the advent of the cyclometer, by the mathematician-cryptologists of the Polish Cipher Bureau (''Biuro Szyfrów'') in decrypting German Enigma machine ciphers. The Enigma rotor cipher machine changes plaintext characters into cipher text using a different permutation for each character, and so implements a polyalphabetic substitution cipher. Background The German navy started using Enigma machines in 1926; it was called ''Funkschlüssel C'' ("Radio cipher C"). By 15 July 1928, the German Army (''Reichswehr'') had introduced their own version of the Enigma—the ''Enigma G''; a revised ''Enigma I'' (with plugboard) appeared in June 1930. The Enigma I used by the German military in the 1930s was a 3-rotor machine. Initially, there were only three rotors labeled ''I'', ''II'', and ''III'', but they could be arranged in any order when placed in the machine. Rejewski identified the rotor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He was one of the first to state and rigorously prove theorems of calculus, rejecting the heuristic principle of the generality of algebra of earlier authors. He almost singlehandedly founded complex analysis and the study of permutation groups in abstract algebra. A profound mathematician, Cauchy had a great influence over his contemporaries and successors; Hans Freudenthal stated: "More concepts and theorems have been named for Cauchy than for any other mathematician (in elasticity alone there are sixteen concepts and theorems named for Cauchy)." Cauchy was a prolific writer; he wrote approximately eight hundred research articles and five complete textbooks on a variety of topics in the fields of mathematics and mathematical physics. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polish Enigma Doubles
A Polish Enigma "double" was a machine produced by the Polish Cipher Bureau that replicated the German Enigma rotor cipher machine. The Enigma double was one result of Marian Rejewski's remarkable achievement of determining the wiring of the Enigma's rotors and reflectors. First double The Polish Cipher Bureau recognized that the Germans were using a new cipher. The Germans had mistakenly shipped a cipher machine to Poland; their attempts to recover a shipment raised the suspicions of Polish customs, and the Polish Cipher Bureau learned that the Germans were using an Enigma machine. The Bureau purchased a commercial Enigma machine, and it attempted but failed to break the cipher. In December 1932, the Polish Cipher Bureau tasked Marian Rejewski Marian Adam Rejewski (; 16 August 1905 – 13 February 1980) was a Polish mathematician and cryptologist who in late 1932 reconstructed the sight-unseen German military Enigma cipher machine, aided by limited documents obtaine ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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