In either case, we wind up with z.
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Homomorphic cryptography offers a similar pair of pathways. We can do arithmetic directly on the plaintext inputs x and y. Or we can encrypt x and y , apply a series of operations to the ciphertext values, then decrypt the result to arrive at the same final answer.
The two routes pass through parallel universes: Arithmetic in plainspace is familiar to everyone. A number is conveniently represented as a sequence of bits binary digits 0 and 1 and algorithms act on the bits according to rules of logic and arithmetic.
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Among the many operations on numbers we might consider, it turns out that adding and multiplying are all we really need to do; other computations can be expressed in terms of these primitives. Doing mathematics in cipherspace is much stranger. Indeed, the task seems all but impossible. Encryption is a process that thoroughly scrambles the bits of a number, whereas algorithms for arithmetic are extremely finicky and give correct results only if all the bits are in the right places. Nevertheless, it can be done. As a proof of concept, I offer an extremely simple homomorphic cryptosystem.
Assume the plaintext consists of integers. To encrypt a number, double it; to decrypt, divide by 2. With this scheme we can do addition on enciphered data as well as a slightly nonstandard version of multiplication. Given plaintext inputs x and y , we can encrypt each of them separately, add the ciphertexts, then decrypt the result. Fiddling with definitions in order to get the right answer may seem like cheating, but many mathematical objects come with their own idiosyncratic rules for multiplication. Two examples are matrices and complex numbers.
We can do all the arithmetic we want on ciphertexts. On the other hand, the system is not recommended if you actually want to keep secrets. Doubling a number does not thoroughly scramble the bits; it merely shifts them left by one position. Devising a secure fully homomorphic cryptosystem is much harder. Making the system efficient enough for practical applications is yet another challenge, still being addressed. Dertouzos, who were all then at MIT. Just a few months before, Rivest and Adleman, along with Adi Shamir, had introduced the first implementation of a public-key cryptosystem, which came to be known as RSA after their initials.
The RSA paper, by the way, also introduced Alice and Bob in their debut performance as celebrity cryptographers. The basic RSA scheme is partially homomorphic: It allows multiplication of ciphertexts but not addition. In the next 30 years there were occasional advances on this front. For example, in Dan Boneh, Eu-Jin Goh and Kobbi Nissim devised a homomorphic system that allowed an unlimited number of additions on the ciphertext, followed by a single multiplication.
He creates a cryptosystem with the usual encrypt and decrypt functions, which convert bits from plaintext to ciphertext and back. He also builds an evaluate function that accepts a description of a computation to be performed on the ciphertext. The computation is specified not as a sequential program but as a circuit or network, where input signals pass through a cascade of logic gates. The evaluate function amounts to a complete computer embedded in the cryptosystem.
In principle, it can calculate any computable function, provided that the circuit representing the function is allowed to extend to arbitrary depth. The depth of a circuit is the number of gates on the longest path from input to output. A full-powered computer must be able to handle circuits of arbitrary depth. Here the homomorphic system runs into a barrier.
Every arithmetic operation amplifies the noise, until eventually it overwhelms the signal. Encrypted data can be envisioned as points purple disks that are given small random displacements from a finite set of lattice points white disks. On decryption, each purple disk is attracted to the nearest white lattice point.
Homomorphic operations amplify the random displacements. If the noise level exceeds a threshold, some of the disks gravitate to the wrong lattice point, leading to an incorrect decryption red arrows. Without some means of noise control, the system can support only a limited number of homomorphic operations. The origin of the noise lies in the probabilistic encryption process. Think of each ciphertext value as a point in space. The decrypt function filters out the noise by treating each point as if it were located at the nearest unperturbed position. When the noise is amplified by homomorphic computations, however, the point wanders farther from its correct position, until finally the decrypt function will associate it with an incorrect plaintext value.
Roughly speaking, each homomorphic addition doubles the noise, and each multiplication squares it. Hence the number of operations must be limited or errors will accumulate. The depth limit could be evaded in the following way: Whenever the noise begins to approach the critical threshold, decrypt the data and then re-encrypt it, thereby resetting the noise to its original low level. The trouble is, decryption requires the secret key, and the whole point of FHE is to allow computation in a context where that key is unavailable. This is where the story gets wacky and wonderful.
The evaluate function built into the cryptosystem is capable of performing any computation, provided it does not exceed the noise limit on circuit depth. So we can ask evaluate to run the decrypt function. Evaluate is designed to work with encrypted data, so the secret key supplied to it in this circumstance is an encrypted version of the normal key; specifically, the secret key supplied to decrypt running within evaluate is the ciphertext produced when encrypt is applied to the plaintext of the secret key.
When decrypt is run with this enciphered key, the result is not plaintext but a new encryption of the ciphertext, with reduced noise. A noise-abatement mechanism was invented by Craig Gentry in But decryption requires a secret key, which is not available. The solution is to run the ciphertext through the decryption algorithm, but with an encrypted version of the decryption key.
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The result is a new ciphertext, just as secure as the original but with lower noise. Here the refreshed ciphertext is represented by numerals from another alphabet, this one Arabic. In effect, Alice is giving Bob a copy of the key needed to unlock the data, but the key is inside a securely locked box and can only be used within that box. As a matter of fact, the box is locked with the very key that is locked inside the box!
Gentry discusses an even more elaborate version of this dizzying metaphor, in which Alice manufactures jewelry in the locked boxes.
Alice in Wonderland
The pause to re-encrypt and refresh the noisy ciphertext can be repeated as needed. In this way the computer can handle a circuit of any finite depth, and the system becomes fully homomorphic. It can carry out arbitrarily complex computations on encrypted data. An essential assumption in this scheme is that the decrypt circuit is itself shallow enough to run without exceeding the noise threshold. From Wikipedia, the free encyclopedia. For the novel that inspired the movie, see Practical Magic novel.
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Magic Box Minimates Alice In Wonderland Through The Looking Glass | litlgeeks
Sandra Bullock as Sally Owens, a witch who becomes widowed after the Owens curse kills her husband. Camilla Belle as young Sally Nicole Kidman as Gillian Owens, Sally's sister, who grows restless with her family's estrangement in their small town, leaves, and becomes the victim of an abusive relationship. Lora Anne Criswell as young Gillian Goran Visnjic as James 'Jimmy' Angelov, Gillian's lover, who becomes abusive and kidnaps the sisters, and is killed by them, twice, in self-defense.
Caprice Benedetti as Maria Owens, the first witch in the Owens family. Evan Rachel Wood as Kylie Owens, Sally's elder daughter, who lives with her mom and great-aunts after the death of her father, Michael. She looks and acts a lot like her Aunt Gillian. Alexandra Artrip as Antonia Owens, Sally's younger daughter, who also lives with her mom and great-aunts after the death of her father, Michael. She looks a lot like her mother. He is a victim of the "Owens Curse", which results in his death. Peter Shaw as Jack, Sally and Gillian's father, who died from the Owens curse when they were children.
Caralyn Kozlowski as Regina, Sally and Gillian's mother, who died of a broken heart after losing her husband to the Owens curse. Chloe Webb as Carla, a friend of Sally's, who works at her shop. Lucinda Jenney as Sara, one of the town women, who initially fears the Owens but later responds to Sally's call for help. Margo Martindale as Linda Bennett, one of town women who responds to Sally's call for help.
Martha Gehman as Patty, one of town women who responds to Sally's call for help. A Victorian House Fit for a Witch". Retrieved 31 October Archived from the original on 6 May Though this Victorian house looks as if it's been in place for a century, it's actually an architectural shell. Retrieved December 4, Retrieved December 18, Films directed by Griffin Dunne. Works by Akiva Goldsman.
Winter's Tale Stephanie Insurgent The 5th Wave Rings Transformers: Legend of the Sword The Dark Tower Retrieved from " https: Articles needing additional references from November All articles needing additional references All articles with unsourced statements Articles with unsourced statements from August Articles with hAudio microformats. Views Read Edit View history. In other projects Wikiquote. This page was last edited on 5 November , at Alice disappears and we land firmly back in contemporary Berlin. Please also see Ethel , Al , Turing Machine. Magic Box melds 19th century illusionism with current technologies in the guise of a pre-cinema device.
The box hangs from the ceiling with brass viewing tubes on either end. Peering inside the device, each participant watches a film that is completely invisible to the other viewer. At the same time they can see right through the box into each others eyes. Levers on the side of the box allow the participants to select which of four films they will view. If both move to the end position, their hands will touch.