en>PhysicsAboveAll at 10:46, 11 January 2014

2014-01-11T10:46:29Z

← Older revision		Revision as of 12:46, 11 January 2014
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	~~Andrew Berryhill is what his spouse loves to contact him and he completely digs that title. Doing ballet is something she would never~~ [http://www.~~indosfriends~~.~~com~~/~~profile~~-~~253~~/~~info~~/ ~~love psychic~~] ~~give up~~. ~~Office supervising~~ is ~~where her main income arrives from but she~~'s ~~currently utilized for an additional one~~. ~~Mississippi~~ is ~~exactly where her home~~ is ~~but her spouse wants them~~ to ~~transfer~~.<br><br>~~My blog~~ ... ~~email~~ [~~http:~~//~~netwk~~.~~hannam~~.ac.kr/xe/~~data_2~~/~~85669 love psychic readings] readings~~ (~~[http~~://~~www~~.~~prayerarmor~~.~~com~~/~~uncategorized~~/~~dont-know-which-kind~~-of~~-hobby-to-take-up-read-~~the-following~~-tips~~/ ~~www~~.~~prayerarmor~~.~~com~~])		The '''Damgård–Jurik cryptosystem'''<ref>[[Ivan Damgård]], Mads Jurik: [http://www.brics.dk/RS/00/45/ A Generalisation, a Simplification and Some Applications of Paillier's Probabilistic Public-Key System]. Public Key Cryptography 2001: 119-136</ref> is a generalization of the [[Paillier cryptosystem]]. It uses computations modulo <math>n^{s+1}</math> where <math>n</math> is an [[RSA (algorithm)\|RSA]] modulus and <math>s</math> a (positive) [[natural number]]. Paillier's scheme is the special case with <math>s=1</math>. The order <math>\varphi(n^{s+1})</math> ([[Euler's totient function]]) of <math>Z^_{n^{s+1}}</math> can be divided by <math>n^s</math>. Moreover <math>Z^_{n^{s+1}}</math> can be written as the [[direct product]] of <math>G \times H</math>. <math>G</math> is cyclic and of order <math>n^s</math>, while <math>H</math> is isomorphic to <math>Z^*_n</math>. For encryption, the message is transformed into the corresponding [[coset]] of the [[factor group]] <math>G/H</math> and the security of the scheme relies on the difficulty of distinguishing random elements in different cosets of <math>H</math>. It is [[semantically secure]] if it is hard to decide if two given elements are in the same coset. Like Paillier, the security of Damgård–Jurik can be proven under the [[decisional composite residuosity assumption]].

			== Key generation ==
			#Choose two large [[prime number]]s ''p'' and ''q'' randomly and independently of each other.
			#Compute <math>n=pq</math> and <math>\lambda=\operatorname{lcm}(p-1,q-1)</math>.
			#Choose an element <math>g \in \mathbb{Z}^*_{n^{s+1}}</math> such that <math>g=(1+n)^j x \mod n^{s+1}</math> for a known <math>j</math> [[relative prime]] to <math>n</math> and <math>x \in H</math>.
			#Using the [[Chinese Remainder Theorem]], choose <math>d</math> such that <math>d \mod n \in \mathbb{Z}^*_n</math> and <math>d= 0 \mod \lambda</math>. For instance <math>d</math> could be <math>\lambda</math> as in Paillier's original scheme.
			*The public (encryption) key is <math>(n, g)</math>.
			*The private (decryption) key is <math>d</math>.

			== Encryption ==
			#Let <math>m</math> be a message to be encrypted where <math>m\in \mathbb Z_{n^s}</math>.
			#Select random <math>r</math> where <math>r\in \mathbb Z^{*}_{n^{s+1}} </math>.
			#Compute ciphertext as: <math> c=g^m \cdot r^{n^s} \mod n^{s+1} </math>.

			== Decryption ==
			#Ciphertext <math>c\in \mathbb Z^{*}_{n^{s+1}} </math>
			#Compute <math>c^d \;mod\;n^{s+1}</math>. If ''c'' is a valid ciphertext then <math>c^d = (g^m r^{n^s})^d = ((1+n)^{jm}x^m r^{n^s})^d = (1+n)^{jmd \;mod\; n^s} (x^m r^{n^s})^{d \;mod\; \lambda} = (1+n)^{jmd \;mod\; n^s}</math>.
			#Apply a recursive version of the Paillier decryption mechanism to obtain <math>jmd</math>. As <math>jd</math> is known, it is possible to compute <math>m=(jmd)\cdot (jd)^{-1} \;mod\;n^s</math>.

			== Simplification ==
			At the cost of no longer containing the classical [[Paillier cryptosystem]] as an instance, Damgård–Jurik can be simplified in the following way:
			*The base ''g'' is fixed as <math>g=n+1</math>.
			*The decryption exponent ''d'' is computed such that <math>d=1 \;mod\; n^s</math> and <math>d=0 \;mod\; \lambda</math>.

			In this case decryption produces <math>c^d = (1+n)^{m} \;mod\; n^{s+1}</math>. Using recursive Paillier decryption this gives us directly the plaintext ''m''.

			==See also==
			* The [http://security.hsr.ch/msevote/damgardjurik Damgård–Jurik cryptosystem interactive simulator] demonstrates a voting application.

			==References==
			{{Reflist}}

			{{Cryptography navbox \| public-key}}

			{{DEFAULTSORT:Damgard-Jurik Cryptosystem}}
			[[Category:Public-key encryption schemes]]

en>BattyBot: General fixes, removed orphan tag using AWB (7961)

2012-02-17T09:37:17Z

General fixes, removed orphan tag using AWB (7961)

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en>PhysicsAboveAll at 10:46, 11 January 2014

en>BattyBot: General fixes, removed orphan tag using AWB (7961)