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	Definition of Trapdoor function
		A trapdoor function is a function that is easy to compute in one direction, and difficult to compute in the opposite direction (finding its inverse) without special information, called the "trapdoor." Trapdoor functions are widely used in cryptography.
		Trapdoor functions came to prominence in cryptography in the mid-1970s with the publication of asymmetric encryption techniques by Diffie, Hellman, and Merkle.
		Functions related to the hardness of the discrete logarithm problem (either modulo a prime or in a group defined over an elliptic curve) are not known to be trapdoor functions, because there is no known "trapdoor" information about the group that enables the efficient computation of discrete logs.
	www.wordiq.com /definition/Trapdoor_function (389 words)

	Trapdoor function - Wikipedia, the free encyclopedia
		Trapdoor functions came to prominence in cryptography in the mid-1970s with the publication of asymmetric encryption techniques by Diffie, Hellman, and Merkle.
		Functions related to the hardness of the discrete logarithm problem (either modulo a prime or in a group defined over an elliptic curve) are not known to be trapdoor functions, because there is no known "trapdoor" information about the group that enables the efficient computation of discrete logs.
		A trapdoor in cryptography has the very specific aforementioned meaning and is not to be confused with a backdoor (these are frequently used interchangeably and this is incorrect).
	en.wikipedia.org /wiki/Trapdoor_function (459 words)

	Trapdoor - Wikipedia, the free encyclopedia
		Hidden trapdoors occasionally appear in fiction, either as entrances to secret passageways, or as literal traps into which a hapless pedestrian may fall if he or she happens to stand on one.
		A trapdoor function is a mathematical function that is easy to compute in one direction, yet (believed to be) difficult to compute in the opposite direction (finding its inverse) without special information, called the "trapdoor".
		Trapdoor spiders are a family of spiders that construct burrows with trapdoor-like entrances.
	en.wikipedia.org /wiki/Trapdoor (243 words)

	MD5z.com :: Encryption and Security Digest
		Hash functions that are truly random with uniform output (including most cryptographic hash functions) are good in that, on average, only one or two probes will be needed (depending on the load factor).
		Aside from minimizing collisions, the hash function for a hash table should also be fast relative to the cost of retrieving a record in the table, as the goal of minimizing collisions is minimizing the time needed to retrieve a desired record.
		The hash function is computed for the data at the sender, and the value of this hash is sent with the data.
	www.md5z.com (1334 words)

	RoboRally.Com - Elements (Q-Z) (Site not responding. Last check: 2007-10-14)
		Function: Robots moving into or through a Radio Beam receive a remote move, which they have to use in their program for the next turn.
		Function: A robots ending their move on a Repeater must execute its current program card once again, if it is a MOVEMENT card (Move 1, 2, 3 or Back up).
		Function: Every robot attempting to end its move on the sand slips down one square (without changing the direction he is looking in) In the corners of the crater, the robot slips down diagonaly.
	www.roborally.com /elements_q.htm (1187 words)

	My Trapdoor Carbine "Shootin' Iron"
		The "Trapdoor" Carbine shown above was assembled by the author during the early to mid 1970's, from the chopped remains of an "after-market" trapdoor rifle generically known as a "Bannerman Special".
		Such firearms were assembled by Francis Bannerman and Sons, and other, less well-known vendors, from earlier-revision trapdoor parts that the U.S. Arsenal at Springfield, Massachusetts, had sold as "scrap metal"; scrap metal is generally melted down for recycling.
		A complete manual on how to load cartridges for original trapdoor rifles and carbines (this applies to all old trapdoors, arsenal or vendor made, as long as they have the original, deep-groove government barrels) is available from Wolf's Western Traders.
	www.sonic.net /~johnpipe/trapdoor (1374 words)

	IEEE P1363: Protocols from other families of public-key algorithms
		EPOC-2 is a public-key encryption system that uses a one-way trapdoor function, two random functions (hash functions) and a symmetric-key encryption (e.g., one-time padding and block-ciphers).
		The trapdoor technique with EPOC is fundamentally different from any other previous scheme including RSA-Rabin and Diffie-Hellman-ElGamal.
		The encryption scheme described in this contribution is obtained by combining three results: one [25] on the trapdoor function technique is by Okamoto and Uchiyama, and the others [13, 14] on conversion techniques using random functions are by Fujisaki and Okamoto.
	grouper.ieee.org /groups/1363/StudyGroup/NewFam.html (925 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		The same function in a C program might look like: int F(int x) { return 7 * x + 43; } Of course, in a computer program, functions are used to encapsulate all kinds of algorithms, and frequently make use of external variables and the like.
		This, finally, is the heart of what makes RSA a trapdoor function: the gap between obtaining a number with two prime factors, and rediscovering the factors from the number itself.
		Gardner, Martin: "Penrose Tiles to Trapdoor Ciphers", 1989, W.H. Freeman & Co. (This is another anthology of Gardner's wonderful columns for "Scientific American", and includes the column which was the first widely published description of the RSA cipher -- the one which set the NSA to frantically running around in circles.) 3.
	www.zone-h.org /files/33/how_rsaw.txt (5399 words)

	Quantum computing: A two-edged sword for security (Site not responding. Last check: 2007-10-14)
		Simply put, a trapdoor function is a calculation that is easy to perform one way but very hard to reverse.
		One such function is multiplying two humongous prime numbers to yield a third humongous number.
		Weirdly, at the same time that quantum computing is shedding doubts on the safety of trapdoor based encryption it is simultaneously solving one of the biggest deployment problems with trapdoor based encryption.
	www.propylon.com /news/ctoarticles/040720_quantum.html (595 words)

	[No title]
		The same function in a C program might look like: int F(int x) { return 7 * x + 43; } Of course, in a computer program, functions are used to encapsulate all kinds of algorithms, and frequently make use of external variables and the like.
		There are some special functions that, though what they do is simple enough, and how they do what they do is utterly transparent, figuring out how to undo what they do is a diabolical task.
		This, finally, is the heart of what makes RSA a trapdoor function: the gap between obtaining a number with two prime factors, and rediscovering the factors from the number itself.
	www.comms.scitech.susx.ac.uk /fft/crypto/the_math_behind_the_rsa_cipher.txt (5400 words)

	Knapsack Cryptosystems: The Past and the Future
		Inspired by the idea of public key cryptography and trapdoor one-way function, Ronald L. Rivest, Adi Shamir and Leonard M. Adleman [80] invented the first public-key cryptosystem, which is based on integer factorization, but not on knapsacks.
		As the trapdoor one-way function is the soul of a public-key cryptosystem, it is essential to understand the trapdoor one-way function used in a knapsack cryptosystem.
		In 2000, IEEE adopted P1363: Standard Specifications For Public Key Cryptography [34], in which three families of cryptographic functions (discrete logarithm in the group of remainders modulo a prime, discrete logarithm in the group of points on an elliptic curve over a finite field, and integer factorization) are discussed.
	www.ics.uci.edu /~mingl/knapsack.html (8138 words)

	The Theory of Cryptography > Public-Key Cryptography (Site not responding. Last check: 2007-10-14)
		Factoring large numbers is a one-way trapdoor function, whereas computing discrete logarithms is a one-way function with no trapdoors.
		The most famous of the well-known trapdoor one-way functions is based on the ease of multiplying two large prime numbers; the reverse process, factoring a very large number, is far more complex.
		Inverting the one-way functions we have discussed, namely, factoring a very large number and computing the discrete logarithm, is known to be practically infeasible within the computing means and the theoretic knowledge available today.
	www.awprofessional.com /articles/article.asp?p=170808&seqNum=3 (5548 words)

	perl.com: Asymmetric Cryptography in Perl
		Central to asymmetric cryptography, is the notion of trapdoor one-way functions.
		Trapdoor one-way functions are much like one-way functions except they are easy to reverse if certain ``trap-door'' information is available.
		The encryption operation consists of computing the trapdoor one-way function on the message (in the easy direction), using the public key, to generate the ciphertext.
	www.perl.com /pub/a/2001/09/26/crypto1.html (1166 words)

	[No title]
		These functions may be thought of as abuscation of point-functions (under a computational assumptions).
		The functions which take on a random value at one (secret) location and are 0 everywhere else.
		Namely, the order of quantifiers is that first fix cryptographic scheme in ROM, then you choose random function in F and give its code to all the users to use instead of H when implementing the cryptographic scheme.
	theory.lcs.mit.edu /classes/6.885/spring05/lecture-notes/lecture7-9 (2409 words)

	Public-Key Cryptography - Quantum Information Technology Group
		A postbox with a one-way chute and a trapdoor that opens with a secret key is implemented mathematically by a trapdoor (one-way) function, i.e.
		The output of the function is the ciphertext.
		This trapdoor function f is the basis for the RSA Cryptosystem: (e,pq) is the public encryption key and (p,q,d) is the secret decryption key.
	www.quantumlah.org /tutorial/publickey (999 words)

	22C:116, Lecture 37, Spring 1999
		The standard function f is a trapdoor function, that is, a function which is very easy to compute in the forward direction but with a very difficult to compute inverse.
		The use of a trapdoor function to verify capabilities means that only the server may restrict the rights on a capability.
		Because there is a standard one-way function available to all users, a user of a capability with a rights mask of all ones may restrict it locally.
	www.cs.uiowa.edu /~jones/opsys/spring99/notes/37.html (1429 words)

	Part Three: Breaking The Public Key Cryptosystems
		The front end function of all the modern PKC's utilize the RSA algorithm for authentication, signatures, and passing keys; however long messages are encrypted with a block cipher such as DES[1] or IDEA[20].
		When functions are implemented in the CORDIC algorithm, after a scaling multiply, all that is required is successive shift and add operations, with each addition supplying an additional digit of precision.
		We will assume for estimating purposes that Schroeppel's algorithm requires the evaluation of 5 elementary functions (this is a guess), and must be iterated 10 times for convergence to the right answer (these numbers could be low by 2 or three orders of magnitude and the point of this discussion would not change).
	www.ee.ualberta.ca /courses/ee401/microboard/cordic_keybreak.html (2298 words)

	No title
		Still, trapdoor function families can be used as the basis for the construction of cryptosystems and digital signature schemes satisfying strong notions of security.
		The RSA function is still the most widely used cryptographic function today and had an enormous impact in practice.
		Getting trapdoor permutation families based on the hardness of the DH problem is an open problem in cryptography.
	www-cse.ucsd.edu /classes/sp05/cse208/lec-dhrsa.html (1674 words)

	The Theory of Cryptography > Public-Key Cryptography (Site not responding. Last check: 2007-10-14)
		The most famous of the well-known trapdoor one-way functions is based on the ease of multiplying two large prime numbers; the reverse process, factoring a very large number, is far more complex.
		Generally speaking, the strength of each algorithm is directly related to the type of the one-way function being used and the length of the cryptographic keys.
		Inverting the one-way functions we have discussed, namely, factoring a very large number and computing the discrete logarithm, is known to be practically infeasible within the computing means and the theoretic knowledge available today.
	www.informit.com /articles/article.asp?p=170808&seqNum=3 (5536 words)

	G12FCO -- Formal Computation, lecture 14
		It is an abuse of notation, but only a small one, to use the same notations to describe functions which generate keys as their results.
		We might like our candidate trapdoor function to have an NP-complete inverse, for the NP-complete problems are, in some sense, the hardest in NP, but there are theoretical reasons why this is actually unlikely.
		Finally, thinking of a clever trapdoor function f is insufficient; we also need to design tractable algorithms D and E such that D (d, E (f(d), m)) = m for all m.
	www.maths.nott.ac.uk /personal/anw/G12FCO/lect14.html (1418 words)

	[No title]
		F(x1,, xm) is a function, which we want to compute in such a way, that no information about the inputs is revealed by the computation to the players.
		OT with one-way trapdoor function Intuitively one-way function is a function that easy to compute, but difficult to invert.
		But it doesn’t guarantee that the inputs of the players lay in the domain of the function F. For example, the legal values for Boolean functions are false and true, which are usually represented as 0 and 1 respectively.
	www.cs.huji.ac.il /~ns/SMPC.doc (4270 words)

	[No title]
		In mathematics, however, a function is used solely for the number it returns.
		Given such a function, it is always possible to construct its inverse.
		And in 1977, Ronald L. Rivest, Adi Shamir, and Leonard Adleman outlined, in an MIT technical memo, an excellent candidate that became the basis for the RSA cipher.
	forums.devshed.com /attachment.php?attachmentid=6951 (2029 words)

	Cryptography-Digest Digest #271
		In a classic cryptosystem, we have encryption functions E_K and decryption functions D_K such that D_K(E_K(P)) = P for any plaintext P. In a public-key cryptosystem, E_K can be easily computed from some ``public key'' X which in turn is computed from K. X is published, so that anyone can encrypt messages.
		Intrinsic to public key cryptography is a `trapdoor function' D_K with the properties that computation in one direction (encryption, E_K) is easy and in the other is virtually impossible (attack, determining P from encryption E_K(P) and public key X).
		Digital signature functions have private hashing and public verification: only one person can produce the hash for a message, but everyone can verify that the hash is correct.
	www.mail-archive.com /cryptography-digest@senator-bedfellow.mit.edu/msg01477.html (2335 words)

	C++ Home
		Of course, in a computer program, functions are used to encapsulate all kinds of algorithms, and frequently make use of external variables and the like.
		Usually, when you have a mathematical function for which an inverse does exist, constructing it is not too difficult.
		In 1975, Whitfield Diffie, Martin E. Hellman, and Ralph Merkle realized that a trapdoor function could be the basis for an entirely new kind of cipher -- one in which the decoding method could remain secret even when the encoding method was public knowledge.
	www.cpp-home.com /tutorial.php?139_1 (2066 words)

	Mathematical Games
		Not only that, but if it were to become necessary at some future time to prove to a third party, say a judge in a court of law, that A did in fact send the message, this can be done in a way that neither A, Z nor anyone else can dispute.
		Diffie and Hellman are applying for patents on cipher devices based on trapdoor functions they have not yet disclosed.
		Since World War II even those government and military ciphers that are not one-time pads have become so difficult to break that the talents of these experts have gradually become less useful.
	www.fortunecity.com /emachines/e11/86/cipher1.html (3343 words)

	sci.crypt: Re: WANTED: trapdoor one-way function
		Sorry, this is not a trapdoor one-way function, it's only a one-way function.
		I think that a *trapdoor* one-way function that satisfies all of your criteria
		trapdoor information is used as the master key.
	www.derkeiler.com /Newsgroups/sci.crypt/2003-07/1004.html (277 words)

	Abstract (Site not responding. Last check: 2007-10-14)
		A basic building block in public key cryptography are the one-way trapdoor functions.
		The best known one-way trapdoor function is the RSA function whose difficulty of inverting is related to the difficulty of factoring.
		Other one-way trapdoor functions use the arithmetic of elliptic curves and more general abelian varieties.
	www.nd.edu /~magic05/abstracts/rosenthal.html (196 words)

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