Benaloh cryptosystem - Revision history

en>Factorize: fixed et al. punctuation

2015-01-06T19:44:12Z

fixed et al. punctuation

← Older revision		Revision as of 21:44, 6 January 2015
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	~~Note: this~~ is ~~not~~ to ~~be confused with the [[Naccache–Stern knapsack cryptosystem]].~~		Alyson is what my husband loves to call me but I don't like when individuals use my full name. For many years she's been working as a travel agent. What me and my family members adore is bungee jumping but I've been taking on psychic readings [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 online psychic readings] ([http://www.seekavideo.com/playlist/2199/video/ www.seekavideo.com]) new issues lately. My spouse and I reside in Mississippi but now I'm contemplating other options.<br><br>My web-site - accurate psychic readings ([http://www.january-yjm.com/xe/index.php?mid=video&document_srl=158289 made my day])

	~~The '''Naccache–Stern cryptosystem''~~' is a [[Homomorphic encryption\|homomorphic]] [[public-key cryptosystem]] whose security rests on the [[higher residuosity problem]]. The Naccache–Stern cryptosystem was discovered by [[David Naccache]] and [[Jacques Stern]] in 1998.

	~~==Scheme Definition==~~

	~~Like~~ many ~~[[Public key cryptography\|public key cryptosystems]], this scheme works in the group <math>(\mathbb{Z}/n\mathbb{Z})^*</math> where~~ '~~'n'' is~~ a ~~product of two large [[Prime number\|primes]]~~. ~~This scheme is [[Homomorphic encryption\|homomorphic]]~~ and ~~hence [[Malleability (cryptography)\|malleable]].~~

	~~===Key Generation===~~

	*Pick a family of '~~'k'' small distinct~~ [~~[Prime number\|primes]] ''p''<sub>1<~~/~~sub>,...,''p''<sub>k<~~/~~sub>~~.
	*Divide the set in half and set <math>u = \prod_{i=1}^{k/2} p_i</~~math> and <math>v = \prod_{k/2+1}^k p_i</math>~~.
	*Set <math>\sigma = ~~uv = \prod_{i=1}^k p_i</math>~~
	*Choose large primes ''a'' and ''b'' such that both ''p'' = ~~2''au''+1 and ''q''~~=~~2''bv''+1 are prime.~~
	*Set ''n''=''pq''.
	*Choose a random ''g'' mod ''n'' such that ''g'' has order φ(~~''n'')~~/4.

	~~The public key is the numbers σ,''n'',''g'' and the private key is the pair ''p'',''q''.~~

	~~When ''k''=1 this is essentially the [[Benaloh cryptosystem]]~~.

	~~===Message Encryption===~~
	~~This system allows encryption of a message ''m'' in the group <math>\mathbb{Z}~~/~~\sigma\mathbb{Z}<~~/~~math>.~~

	*Pick a random <math>x \in \mathbb{Z}/~~n\mathbb{Z}<~~/~~math>~~.
	*Calculate <math>E(m) = x^\sigma g^m \mod n</math>

	~~Then ''E(m)'' is an encryption of the message ''m''~~.

	~~===Message Decryption===~~

	~~To decrypt, we first find ''m'' mod ''p''<sub>''i''</sub> for each ''i'', and then we apply the [[Chinese remainder theorem~~]~~] to calculate ''m'' mod <math>\sigma</math>.~~

	~~Given a ciphertext ''c'', to decrypt, we calculate~~

	*<math>c_i \equiv c^{\phi(n)~~/p_i} \mod n</math>~~. ~~Thus~~
	~~:<math> \begin{matrix} c^{\phi(n)/p_i} &\equiv& x^{\sigma \phi(n)/p_i} g^{m\phi(n)/p_i} \mod n\\ &\equiv& g^{(m_i + y_ip_i)\phi(n)/p_i} \mod n \\ &\equiv& g^{m_i\phi(n)/p_i} \mod n \end{matrix}</math>~~
	~~where <math>m_i \equiv m \mod p_i</math>.~~
	*Since ''p''<sub>''i''</sub> is chosen to be small, ''m~~''<sub>''i''</sub> can be recovered be exhaustive search, i.e~~. ~~by comparing~~ <~~math~~>~~c_i~~<~~/math~~> ~~to <math>g^{j\phi~~(~~n)/p_i}</math> for ''j'' from 1 to ''p''<sub>''i''</sub>-1.~~
	*Once ''m''<sub>''i''</sub> is known for each ''i'', ''m'' can be recovered by a direct application of the Chinese remainder theorem.

	~~==Security==~~
	~~The [[semantic security]] of the Naccache–Stern cryptosystem rests on an extension of the [[quadratic residuosity problem]] known as the [[higher residuosity problem]].~~

	~~==References==~~
	[http://~~citeseer~~.~~ist~~.~~psu.edu~~/~~naccache98new~~.~~html Original paper]~~
	~~{{Cryptography navbox \| public-key}}~~

	~~{{DEFAULTSORT:Naccache-Stern cryptosystem}}~~
	~~[[Category:Public-key encryption schemes]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q816670

2013-03-15T22:30:06Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q816670

← Older revision		Revision as of 00:30, 16 March 2013
Line 1:		Line 1:
	The [~~http:~~//~~formalarmour~~.~~com~~/~~index~~.~~php?do~~=/~~profile-26947~~/~~info~~/ ~~accurate psychic predictions] person who wrote the article is called Jayson Hirano~~ and ~~he completely digs~~ that ~~name~~. ~~Since he was 18 he~~'~~s been working as an info officer but he ideas on changing it~~. ~~clairvoyance~~ ([~~http:~~//~~www~~.~~khuplaza.com~~/~~dent~~/~~14869889 www~~.~~khuplaza~~.~~com~~]~~) What me and my family members love is~~ to ~~climb but I~~'m ~~thinking on beginning some thing new~~. ~~For~~ a ~~whilst I~~'~~ve been in Alaska but I will have~~ to ~~transfer in a yr or two~~.<br><br>~~Feel free~~ to ~~visit my webpage~~ ... ~~free tarot readings (~~[http://~~www~~.~~youronlinepublishers~~.~~com~~/~~authWiki/AdolphvhBladenqq Highly recommended Online site~~])		Note: this is not to be confused with the [[Naccache–Stern knapsack cryptosystem]].

			The '''Naccache–Stern cryptosystem''' is a [[Homomorphic encryption\|homomorphic]] [[public-key cryptosystem]] whose security rests on the [[higher residuosity problem]]. The Naccache–Stern cryptosystem was discovered by [[David Naccache]] and [[Jacques Stern]] in 1998.

			==Scheme Definition==

			Like many [[Public key cryptography\|public key cryptosystems]], this scheme works in the group <math>(\mathbb{Z}/n\mathbb{Z})^*</math> where ''n'' is a product of two large [[Prime number\|primes]]. This scheme is [[Homomorphic encryption\|homomorphic]] and hence [[Malleability (cryptography)\|malleable]].

			===Key Generation===

			*Pick a family of ''k'' small distinct [[Prime number\|primes]] ''p''<sub>1</sub>,...,''p''<sub>k</sub>.
			*Divide the set in half and set <math>u = \prod_{i=1}^{k/2} p_i</math> and <math>v = \prod_{k/2+1}^k p_i</math>.
			*Set <math>\sigma = uv = \prod_{i=1}^k p_i</math>
			*Choose large primes ''a'' and ''b'' such that both ''p'' = 2''au''+1 and ''q''=2''bv''+1 are prime.
			*Set ''n''=''pq''.
			*Choose a random ''g'' mod ''n'' such that ''g'' has order φ(''n'')/4.

			The public key is the numbers σ,''n'',''g'' and the private key is the pair ''p'',''q''.

			When ''k''=1 this is essentially the [[Benaloh cryptosystem]].

			===Message Encryption===
			This system allows encryption of a message ''m'' in the group <math>\mathbb{Z}/\sigma\mathbb{Z}</math>.

			*Pick a random <math>x \in \mathbb{Z}/n\mathbb{Z}</math>.
			*Calculate <math>E(m) = x^\sigma g^m \mod n</math>

			Then ''E(m)'' is an encryption of the message ''m''.

			===Message Decryption===

			To decrypt, we first find ''m'' mod ''p''<sub>''i''</sub> for each ''i'', and then we apply the [[Chinese remainder theorem]] to calculate ''m'' mod <math>\sigma</math>.

			Given a ciphertext ''c'', to decrypt, we calculate

			*<math>c_i \equiv c^{\phi(n)/p_i} \mod n</math>. Thus
			:<math> \begin{matrix} c^{\phi(n)/p_i} &\equiv& x^{\sigma \phi(n)/p_i} g^{m\phi(n)/p_i} \mod n\\ &\equiv& g^{(m_i + y_ip_i)\phi(n)/p_i} \mod n \\ &\equiv& g^{m_i\phi(n)/p_i} \mod n \end{matrix}</math>
			where <math>m_i \equiv m \mod p_i</math>.
			*Since ''p''<sub>''i''</sub> is chosen to be small, ''m''<sub>''i''</sub> can be recovered be exhaustive search, i.e. by comparing <math>c_i</math> to <math>g^{j\phi(n)/p_i}</math> for ''j'' from 1 to ''p''<sub>''i''</sub>-1.
			*Once ''m''<sub>''i''</sub> is known for each ''i'', ''m'' can be recovered by a direct application of the Chinese remainder theorem.

			==Security==
			The [[semantic security]] of the Naccache–Stern cryptosystem rests on an extension of the [[quadratic residuosity problem]] known as the [[higher residuosity problem]].

			==References==
			[http://citeseer.ist.psu.edu/naccache98new.html Original paper]
			{{Cryptography navbox \| public-key}}

			{{DEFAULTSORT:Naccache-Stern cryptosystem}}
			[[Category:Public-key encryption schemes]]

en>Etamatic123: /* References */ link to german version

2012-06-13T11:26:13Z

References: link to german version

New page

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