Good–Turing frequency estimation - Revision history

en>Gadget850: /* References */refbegin, replaced: → {{refend}},
→ {{refbegin}} using AWB

2014-12-31T10:43:16Z

References: refbegin, replaced: </div> → {{refend}}, <div class="references-small"> → {{refbegin}} using AWB

← Older revision		Revision as of 12:43, 31 December 2014
Line 1:		Line 1:
	~~The '''leftover hash lemma''' is a [[lemma (mathematics)\|lemma]] in [[cryptography]] first stated by [[Russell Impagliazzo]], [[Leonid Levin]], and [[Michael Luby]].~~		I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. For years she's been operating as a travel agent. Some time ago he chose to reside in North Carolina and he doesn't plan on altering it. To play lacross is some thing he would never give up.<br><br>my blog post :: online reader - [http://mybrandcp.com/xe/board_XmDx25/107997 mybrandcp.com] -

	~~Imagine that you have a secret [[key (cryptography)\|key]] <math>\scriptstyle X</math> that has <math>\scriptstyle n</math> uniform random [[bit]]s, and you~~ would like to ~~use this secret key~~ to ~~encrypt a message. Unfortunately,~~ you ~~were a bit careless with the key~~, and ~~know that an [[adversary (cryptography)\|adversary]] was able to learn about <math>\scriptstyle~~ t ~~\;<\; n</math> bits of that key, but you do not know which. Can you still use your key, or do you have to throw~~ it away and choose a new key? The leftover hash lemma tells us that we can produce a key of [[almost all\|almost]] <math>\scriptstyle n \,-\, t</math> bits, over which the adversary has almost no knowledge. Since the adversary knows all ~~but <math>\scriptstyle n \,-\, t</math> bits, this is [[almost optimal]]~~.

	More precisely, the leftover hash lemma tells us that we can extract about <math>\scriptstyle H_\infty(X)</math> (the [[min-entropy]] of <math>\scriptstyle X</math>) bits from a [[random variable]] <math>\scriptstyle X</math> that are almost uniformly distributed. In other words, an adversary who has some partial knowledge about <math>\scriptstyle X</math>, will have almost no knowledge about the extracted value. That is why this is also called '''privacy amplification''' ~~(see privacy amplification section in the article [[Quantum key distribution]]).~~

	~~[[Randomness extractor]]~~s ~~achieve the same result, but use (normally) less randomness.~~

	~~==Leftover hash lemma==~~
	~~Let <math>\scriptstyle X</math> be~~ a ~~random variable over <math>\scriptstyle \mathcal X</math> and let <math>\scriptstyle m \;>\; 0</math>~~. ~~Let <math>\scriptstyle h :\; \mathcal{S} \,\times\, \mathcal{X} \;\rightarrow\; \{0,\, 1\}^m</math> be a 2-[[universal hashing\|universal]] [[hash function]]. If~~
	~~:<math>m \leq H_\infty(X) - 2 \log\left(\frac{1}{\varepsilon}\right)</math>~~

	~~then for <math>\scriptstyle S</math> uniform over <math>\scriptstyle \mathcal S</math>~~ and ~~independent of <math>\scriptstyle X</math>, we have~~
	~~:<math>\delta[(h(S, X), S), (U, S)] \leq \varepsilon</math>~~

	~~where <math>\scriptstyle U</math> is uniform over <math>\scriptstyle \{0,\, 1\}^m</math> and independent of <math>\scriptstyle S</math>~~.

	~~<math>\scriptstyle H_\infty(X) \;=\; -\log \max_x \Pr[X=x]</math>~~ is ~~the [[Min-entropy]] of <math>\scriptstyle X</math>, which measures the amount of randomness <math>\scriptstyle X</math> has~~. ~~The min-entropy is always less than or equal to the [[Shannon entropy]]. Note that~~ <~~math~~>~~\scriptstyle \max_x \Pr[X=x]~~<~~/math~~> is the probability of correctly guessing <math>\scriptstyle X</math>. (The best guess is to guess the most probable value.) Therefore, the min-entropy measures how difficult it is to guess <math>\scriptstyle X</math>.

	<math>\scriptstyle \delta(X,\, Y) \;=\; \frac{1}{2} \sum_v \left \| \Pr[X=v] \,-\, \Pr[Y=v] \right \|</math> is a [[statistical distance]] between <math>\scriptstyle X</math> and <math>\scriptstyle Y</math>.

	~~==See also==~~
	* [[Universal hashing]]
	* [[Min-entropy]]
	* [[Rényi entropy]]
	* [[Information theoretic security]]

	~~==References==~~
	*[http:~~//portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=45477 C. H. Bennett, G. Brassard, and J. M. Robert. ''Privacy amplification by public discussion''. SIAM Journal on Computing, 17(2)~~:~~210~~-~~229, 1988.]~~
	*[http://~~portal.acm~~.~~org~~/citation.cfm?coll=GUIDE&dl=GUIDE&id=73009 R. Impagliazzo, L. A. Levin, and M. Luby. ''Pseudo-random generation from one-way functions''. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing (STOC '89), pages 12-24. ACM Press, 1989.]
	*[http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=10153&arnumber=476316&type=ref C. Bennett, G. Brassard, C. Crepeau, and U. Maurer. ''Generalized privacy amplification''. IEEE Transactions on Information Theory, 41, 1995.]
	*[http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=312213 J. Håstad, R. Impagliazzo, L. A. Levin and M. Luby. ''A Pseudorandom Generator from any One-way Function''. SIAM Journal on Computing, v28 n4, pp. 1364-~~1396, 1999.]~~

	~~[[Category:Theory of cryptography]]~~
	~~[[Category:Probability theorems]]~~

en>DarwinPeacock: trying to clarify intro sentence

2013-10-14T19:52:02Z

trying to clarify intro sentence

← Older revision		Revision as of 21:52, 14 October 2013
Line 1:		Line 1:
	~~Hello friend~~. ~~Let me~~ [~~https:~~//~~www~~.~~geico~~.~~com~~/~~getaquote~~/~~auto~~/~~mechanical~~-~~breakdown~~-~~insurance~~/ ~~introduce~~] ~~myself~~. ~~I am Luther~~ [~~http:~~//~~www~~.~~classicguides~~.~~com~~/~~automotive~~/~~extended~~-~~vehicle~~-~~warranty~~-~~plans/ extended auto warranty~~] [http://~~www~~.~~empredators~~.de/~~index~~.~~php~~?~~mod~~=~~users~~&~~action~~=~~view~~&id=~~10734 extended car warranty~~] ~~auto~~ [http://~~www~~.ag.~~state~~.mn.~~us/consumer/cars/mncarlaws/mncarlaws_2~~.~~asp warranty warranty] Aubrey~~. ~~The favorite pastime for him~~ and ~~his children is to generate and now he is trying to make cash with it~~. ~~My job is a manufacturing and distribution officer and I~~'~~m doing pretty good monetarily~~. ~~Her~~ [http://~~www~~.~~bankrate~~.~~com~~/~~finance~~/~~auto/automakers-best-car-warranties-1~~.~~aspx family life~~] ~~in Delaware car warranty but she requirements to transfer because of her family.<br><br>Also visit my website -~~ [http://~~Www.team~~.~~trustfactory~~.ch/~~index~~.~~php~~?~~mod~~=~~users~~&~~action~~=~~view~~&id=~~4832 Www~~.~~team~~.~~trustfactory~~.ch]		The '''leftover hash lemma''' is a [[lemma (mathematics)\|lemma]] in [[cryptography]] first stated by [[Russell Impagliazzo]], [[Leonid Levin]], and [[Michael Luby]].

			Imagine that you have a secret [[key (cryptography)\|key]] <math>\scriptstyle X</math> that has <math>\scriptstyle n</math> uniform random [[bit]]s, and you would like to use this secret key to encrypt a message. Unfortunately, you were a bit careless with the key, and know that an [[adversary (cryptography)\|adversary]] was able to learn about <math>\scriptstyle t \;<\; n</math> bits of that key, but you do not know which. Can you still use your key, or do you have to throw it away and choose a new key? The leftover hash lemma tells us that we can produce a key of [[almost all\|almost]] <math>\scriptstyle n \,-\, t</math> bits, over which the adversary has almost no knowledge. Since the adversary knows all but <math>\scriptstyle n \,-\, t</math> bits, this is [[almost optimal]].

			More precisely, the leftover hash lemma tells us that we can extract about <math>\scriptstyle H_\infty(X)</math> (the [[min-entropy]] of <math>\scriptstyle X</math>) bits from a [[random variable]] <math>\scriptstyle X</math> that are almost uniformly distributed. In other words, an adversary who has some partial knowledge about <math>\scriptstyle X</math>, will have almost no knowledge about the extracted value. That is why this is also called '''privacy amplification''' (see privacy amplification section in the article [[Quantum key distribution]]).

			[[Randomness extractor]]s achieve the same result, but use (normally) less randomness.

			==Leftover hash lemma==
			Let <math>\scriptstyle X</math> be a random variable over <math>\scriptstyle \mathcal X</math> and let <math>\scriptstyle m \;>\; 0</math>. Let <math>\scriptstyle h :\; \mathcal{S} \,\times\, \mathcal{X} \;\rightarrow\; \{0,\, 1\}^m</math> be a 2-[[universal hashing\|universal]] [[hash function]]. If
			:<math>m \leq H_\infty(X) - 2 \log\left(\frac{1}{\varepsilon}\right)</math>

			then for <math>\scriptstyle S</math> uniform over <math>\scriptstyle \mathcal S</math> and independent of <math>\scriptstyle X</math>, we have
			:<math>\delta[(h(S, X), S), (U, S)] \leq \varepsilon</math>

			where <math>\scriptstyle U</math> is uniform over <math>\scriptstyle \{0,\, 1\}^m</math> and independent of <math>\scriptstyle S</math>.

			<math>\scriptstyle H_\infty(X) \;=\; -\log \max_x \Pr[X=x]</math> is the [[Min-entropy]] of <math>\scriptstyle X</math>, which measures the amount of randomness <math>\scriptstyle X</math> has. The min-entropy is always less than or equal to the [[Shannon entropy]]. Note that <math>\scriptstyle \max_x \Pr[X=x]</math> is the probability of correctly guessing <math>\scriptstyle X</math>. (The best guess is to guess the most probable value.) Therefore, the min-entropy measures how difficult it is to guess <math>\scriptstyle X</math>.

			<math>\scriptstyle \delta(X,\, Y) \;=\; \frac{1}{2} \sum_v \left \| \Pr[X=v] \,-\, \Pr[Y=v] \right \|</math> is a [[statistical distance]] between <math>\scriptstyle X</math> and <math>\scriptstyle Y</math>.

			==See also==
			* [[Universal hashing]]
			* [[Min-entropy]]
			* [[Rényi entropy]]
			* [[Information theoretic security]]

			==References==
			*[http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=45477 C. H. Bennett, G. Brassard, and J. M. Robert. ''Privacy amplification by public discussion''. SIAM Journal on Computing, 17(2):210-229, 1988.]
			*[http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=73009 R. Impagliazzo, L. A. Levin, and M. Luby. ''Pseudo-random generation from one-way functions''. In Proceedings of the 21st Annual ACM Symposium on Theory of Computing (STOC '89), pages 12-24. ACM Press, 1989.]
			*[http://ieeexplore.ieee.org/xpls/abs_all.jsp?isnumber=10153&arnumber=476316&type=ref C. Bennett, G. Brassard, C. Crepeau, and U. Maurer. ''Generalized privacy amplification''. IEEE Transactions on Information Theory, 41, 1995.]
			*[http://portal.acm.org/citation.cfm?coll=GUIDE&dl=GUIDE&id=312213 J. Håstad, R. Impagliazzo, L. A. Levin and M. Luby. ''A Pseudorandom Generator from any One-way Function''. SIAM Journal on Computing, v28 n4, pp. 1364-1396, 1999.]

			[[Category:Theory of cryptography]]
			[[Category:Probability theorems]]

174.53.163.119: /* The method */ some small math notation cleanups per WP:MOSMATH

2012-08-09T18:45:02Z

The method: some small math notation cleanups per WP:MOSMATH

New page

Hello friend. Let me [https://www.geico.com/getaquote/auto/mechanical-breakdown-insurance/ introduce] myself. I am Luther [http://www.classicguides.com/automotive/extended-vehicle-warranty-plans/ extended auto warranty] [http://www.empredators.de/index.php?mod=users&action=view&id=10734 extended car warranty] auto [http://www.ag.state.mn.us/consumer/cars/mncarlaws/mncarlaws_2.asp warranty warranty] Aubrey. The favorite pastime for him and his children is to generate and now he is trying to make cash with it. My job is a manufacturing and distribution officer and I'm doing pretty good monetarily. Her [http://www.bankrate.com/finance/auto/automakers-best-car-warranties-1.aspx family life] in Delaware car warranty but she requirements to transfer because of her family.<br><br>Also visit my website - [http://Www.team.trustfactory.ch/index.php?mod=users&action=view&id=4832 Www.team.trustfactory.ch]

Good–Turing frequency estimation - Revision history

en>Gadget850: /* References */refbegin, replaced: → {{refend}}, → {{refbegin}} using AWB

en>DarwinPeacock: trying to clarify intro sentence

174.53.163.119: /* The method */ some small math notation cleanups per WP:MOSMATH

en>Gadget850: /* References */refbegin, replaced: → {{refend}},
→ {{refbegin}} using AWB