Nov 04, 2014 Meet Eddie Woo, the maths teacher you wish you’d had in high school Australian Story - Duration: 28:09. ABC News In-depth Recommended for you. Asymmetric algorithms require the creation of a public key and a private key. The public key can be made public to anyone, while the private key must known only by the party who will decrypt the data encrypted with the public key. This section describes how to generate and manage keys for both symmetric and asymmetric algorithms.
Maze generation algorithms are automated methods for the creation of mazes.
A maze can be generated by starting with a predetermined arrangement of cells (most commonly a rectangular grid but other arrangements are possible) with wall sites between them. This predetermined arrangement can be considered as a connected graph with the edges representing possible wall sites and the nodes representing cells. The purpose of the maze generation algorithm can then be considered to be making a subgraph in which it is challenging to find a route between two particular nodes.
If the subgraph is not connected, then there are regions of the graph that are wasted because they do not contribute to the search space. If the graph contains loops, then there may be multiple paths between the chosen nodes. Because of this, maze generation is often approached as generating a random spanning tree. Loops, which can confound naive maze solvers, may be introduced by adding random edges to the result during the course of the algorithm.
The animation shows the maze generation steps for a graph that is not on a rectangular grid.First, the computer creates a random planar graph Gshown in blue, and its dual Fshown in yellow. Second, computer traverses F using a chosenalgorithm, such as a depth-first search, coloring the path red.During the traversal, whenever a red edge crosses over a blue edge,the blue edge is removed.Finally, when all vertices of F have been visited, F is erasedand two edges from G, one for the entrance and one for the exit, are removed.
This algorithm is a randomized version of the depth-first search algorithm. Frequently implemented with a stack, this approach is one of the simplest ways to generate a maze using a computer. Consider the space for a maze being a large grid of cells (like a large chess board), each cell starting with four walls. Starting from a random cell, the computer then selects a random neighbouring cell that has not yet been visited. The computer removes the wall between the two cells and marks the new cell as visited, and adds it to the stack to facilitate backtracking. The computer continues this process, with a cell that has no unvisited neighbours being considered a dead-end. When at a dead-end it backtracks through the path until it reaches a cell with an unvisited neighbour, continuing the path generation by visiting this new, unvisited cell (creating a new junction). This process continues until every cell has been visited, causing the computer to backtrack all the way back to the beginning cell. We can be sure every cell is visited.
As given above this algorithm involves deep recursion which may cause stack overflow issues on some computer architectures. The algorithm can be rearranged into a loop by storing backtracking information in the maze itself. This also provides a quick way to display a solution, by starting at any given point and backtracking to the beginning.
Mazes generated with a depth-first search have a low branching factor and contain many long corridors, because the algorithm explores as far as possible along each branch before backtracking.
The depth-first search algorithm of maze generation is frequently implemented using backtracking. This can be described with a following recursive routine:
which is invoked once for any initial cell in the area.
/cleanmymacx-key-generator-online-free.html. A disadvantage of this approach is a large depth of recursion – in the worst case, the routine may need to recur on every cell of the area being processed, which may exceed the maximum recursion stack depth in many environments. As a solution, the same bactracking method can be implemented with an explicit stack, which is usually allowed to grow much bigger with no harm.
This algorithm is a randomized version of Kruskal's algorithm.
There are several data structures that can be used to model the sets of cells. An efficient implementation using a disjoint-set data structure can perform each union and find operation on two sets in nearly constant amortized time (specifically, time; for any plausible value of ), so the running time of this algorithm is essentially proportional to the number of walls available to the maze.
It matters little whether the list of walls is initially randomized or if a wall is randomly chosen from a nonrandom list, either way is just as easy to code.
Because the effect of this algorithm is to produce a minimal spanning tree from a graph with equally weighted edges, it tends to produce regular patterns which are fairly easy to solve.
audio dropouts traktor pro 3 This algorithm is a randomized version of Prim's algorithm.
It will usually be relatively easy to find the way to the starting cell, but hard to find the way anywhere else.
Note that simply running classical Prim's on a graph with random edge weights would create mazes stylistically identical to Kruskal's, because they are both minimal spanning tree algorithms. Instead, this algorithm introduces stylistic variation because the edges closer to the starting point have a lower effective weight.
Although the classical Prim's algorithm keeps a list of edges, for maze generation we could instead maintain a list of adjacent cells. If the randomly chosen cell has multiple edges that connect it to the existing maze, select one of these edges at random. This will tend to branch slightly more than the edge-based version above.
All the above algorithms have biases of various sorts: depth-first search is biased toward long corridors, while Kruskal's/Prim's algorithms are biased toward many short dead ends. Wilson's algorithm,[1] on the other hand, generates an unbiased sample from the uniform distribution over all mazes, using loop-erased random walks.
We begin the algorithm by initializing the maze with one cell chosen arbitrarily. Then we start at a new cell chosen arbitrarily, and perform a random walk until we reach a cell already in the maze—however, if at any point the random walk reaches its own path, forming a loop, we erase the loop from the path before proceeding. When the path reaches the maze, we add it to the maze. Then we perform another loop-erased random walk from another arbitrary starting cell, repeating until all cells have been filled.
This procedure remains unbiased no matter which method we use to arbitrarily choose starting cells. So we could always choose the first unfilled cell in (say) left-to-right, top-to-bottom order for simplicity.
| original chamber | division by two walls | holes in walls | continue subdividing. | completed |
|---|---|---|---|---|
step 1 | step 2 | step 3 | step 4 | step 5 |
Mazes can be created with recursive division, an algorithm which works as follows: Begin with the maze's space with no walls. Call this a chamber. Divide the chamber with a randomly positioned wall (or multiple walls) where each wall contains a randomly positioned passage opening within it. Then recursively repeat the process on the subchambers until all chambers are minimum sized. This method results in mazes with long straight walls crossing their space, making it easier to see which areas to avoid.
For example, in a rectangular maze, build at random points two walls that are perpendicular to each other. These two walls divide the large chamber into four smaller chambers separated by four walls. Choose three of the four walls at random, and open a one cell-wide hole at a random point in each of the three. Continue in this manner recursively, until every chamber has a width of one cell in either of the two directions.
Other algorithms exist that require only enough memory to store one line of a 2D maze or one plane of a 3D maze. Eller's algorithm prevents loops by storing which cells in the current line are connected through cells in the previous lines, and never removes walls between any two cells already connected.[2] The Sidewinder algorithm starts with an open passage along the entire the top row, and subsequent rows consist of shorter horizontal passages with one connection to the passage above. The Sidewinder algorithm is trivial to solve from the bottom up because it has no upward dead ends.[3] Given a starting width, both algorithm create perfect mazes of unlimited height.
Most maze generation algorithms require maintaining relationships between cells within it, to ensure the end result will be solvable. Valid simply connected mazes can however be generated by focusing on each cell independently. A binary tree maze is a standard orthogonal maze where each cell always has a passage leading up or leading left, but never both. To create a binary tree maze, for each cell flip a coin to decide whether to add a passage leading up or left. Always pick the same direction for cells on the boundary, and the end result will be a valid simply connected maze that looks like a binary tree, with the upper left corner its root. As with Sidewinder, the binary tree maze has no dead ends in the directions of bias.
A related form of flipping a coin for each cell is to create an image using a random mix of forward slash and backslash characters. This doesn't generate a valid simply connected maze, but rather a selection of closed loops and unicursal passages. (The manual for the Commodore 64 presents a BASIC program using this algorithm, but using PETSCII diagonal line graphic characters instead for a smoother graphic appearance.)
Certain types of cellular automata can be used to generate mazes.[4] Two well-known such cellular automata, Maze and Mazectric, have rulestrings B3/S12345 and B3/S1234.[4] In the former, this means that cells survive from one generation to the next if they have at least one and at most five neighbours. In the latter, this means that cells survive if they have one to four neighbours. If a cell has exactly three neighbours, it is born. It is similar to Conway's Game of Life in that patterns that do not have a living cell adjacent to 1, 4, or 5 other living cells in any generation will behave identically to it.[4] However, for large patterns, it behaves very differently from Life.[4]
For a random starting pattern, these maze-generating cellular automata will evolve into complex mazes with well-defined walls outlining corridors. Mazecetric, which has the rule B3/S1234 has a tendency to generate longer and straighter corridors compared with Maze, with the rule B3/S12345.[4] Since these cellular automaton rules are deterministic, each maze generated is uniquely determined by its random starting pattern. This is a significant drawback since the mazes tend to be relatively predictable.
Like some of the graph-theory based methods described above, these cellular automata typically generate mazes from a single starting pattern; hence it will usually be relatively easy to find the way to the starting cell, but harder to find the way anywhere else.
Symmetric-key algorithms[a] are algorithms for cryptography that use the same cryptographic keys for both encryption of plaintext and decryption of ciphertext. The keys may be identical or there may be a simple transformation to go between the two keys.[1] The keys, in practice, represent a shared secret between two or more parties that can be used to maintain a private information link.[2] This requirement that both parties have access to the secret key is one of the main drawbacks of symmetric key encryption, in comparison to public-key encryption (also known as asymmetric key encryption).[3][4]
Symmetric-key encryption can use either stream ciphers or block ciphers.[5]
Examples of popular symmetric-key algorithms include Twofish, Serpent, AES (Rijndael), Blowfish, CAST5, Kuznyechik, RC4, DES, 3DES, Skipjack, Safer+/++ (Bluetooth), and IDEA.[6]
Symmetric ciphers are commonly used to achieve other cryptographic primitives than just encryption.[citation needed]
Encrypting a message does not guarantee that this message is not changed while encrypted. Hence often a message authentication code is added to a ciphertext to ensure that changes to the ciphertext will be noted by the receiver. Message authentication codes can be constructed from symmetric ciphers (e.g. CBC-MAC).[citation needed]
However, symmetric ciphers cannot be used for non-repudiation purposes except by involving additional parties.[7] See the ISO/IEC 13888-2 standard.
Another application is to build hash functions from block ciphers. See one-way compression function for descriptions of several such methods.
Many modern block ciphers are based on a construction proposed by Horst Feistel. Feistel's construction makes it possible to build invertible functions from other functions that are themselves not invertible.[citation needed]
Symmetric ciphers have historically been susceptible to known-plaintext attacks, chosen-plaintext attacks, differential cryptanalysis and linear cryptanalysis. Careful construction of the functions for each round can greatly reduce the chances of a successful attack.[citation needed]
Symmetric-key algorithms require both the sender and the recipient of a message to have the same secret key.All early cryptographic systems required one of those people to somehow receive a copy of that secret key over a physically secure channel.
Nearly all modern cryptographic systems still use symmetric-key algorithms internally to encrypt the bulk of the messages, but they eliminate the need for a physically secure channel by using Diffie–Hellman key exchange or some other public-key protocol to securely come to agreement on a fresh new secret key for each message (forward secrecy).
When used with asymmetric ciphers for key transfer, pseudorandom key generators are nearly always used to generate the symmetric cipher session keys. However, lack of randomness in those generators or in their initialization vectors is disastrous and has led to cryptanalytic breaks in the past. Therefore, it is essential that an implementation use a source of high entropy for its initialization.[8][9][10]
A reciprocal cipher is a cipher where, just as one enters the plaintext into the cryptography system to get the ciphertext, one could enter the ciphertext into the same place in the system to get the plaintext. A reciprocal cipher is also sometimes referred as self-reciprocal cipher.
Practically all mechanical cipher machines implement a reciprocal cipher, a mathematical involution on each typed-in letter.Instead of designing two kinds of machines, one for encrypting and one for decrypting, all the machines can be identical and can be set up (keyed) the same way.[11]
Examples of reciprocal ciphers include:
Practically all modern ciphers can be classified as either a stream cipher, most of which use a reciprocol XOR cipher combiner, or a block cipher, most of which use use Feistel cipher or Lai–Massey scheme with a reciprocal transformation in each round.