en>JorisvS: /* Galactic interaction */ ce

2014-12-31T13:19:31Z

Galactic interaction: ce

← Older revision		Revision as of 15:19, 31 December 2014
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	~~Andrew Berryhill~~ is ~~what his spouse enjoys to call him~~ and he ~~completely digs that title~~. ~~Invoicing~~ is ~~what~~ I ~~do. For years he~~'s been residing in ~~Mississippi and he doesn~~'~~t strategy on changing it~~. ~~The favorite pastime for him and his children~~ is ~~to perform lacross and he would by no means give it up~~.<br><br>~~my webpage :: online reader (~~[http://~~Kjhkkb~~.~~net~~/xe/~~notice/374835 http://Kjhkkb.net~~])		Alyson Meagher is the title her mothers and fathers gave her but she doesn't like when people use her complete title. Credit authorising is how he tends to make money. Alaska is the only location I've been residing in but now I'm considering other options. What I love performing is football but I don't have the time recently.<br><br>My web site - [http://www.chk.woobi.co.kr/xe/?document_srl=346069 email psychic readings]

en>Charles Edwin Shipp: /* See also */ Cosmology

2014-02-28T09:38:41Z

See also: Cosmology

← Older revision		Revision as of 11:38, 28 February 2014
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	~~The '''Benaloh Cryptosystem''' is an extension of the [[Goldwasser-Micali cryptosystem]] (GM) created in 1994 by Josh (Cohen) Benaloh. The main improvement of the Benaloh Cryptosystem over GM~~ is that ~~longer blocks of data can be encrypted at once, whereas in GM each bit is encrypted individually~~.		Andrew Berryhill is what his spouse enjoys to call him and he completely digs that title. Invoicing is what I do. For years he's been residing in Mississippi and he doesn't strategy on changing it. The favorite pastime for him and his children is to perform lacross and he would by no means give it up.<br><br>my webpage :: online reader ([http://Kjhkkb.net/xe/notice/374835 http://Kjhkkb.net])

	~~==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===~~
	~~A public/private key pair is generated as follows:~~

	*Choose a blocksize ''r''.
	*Choose large primes ''p'' and '~~'q'' such that ''r'' divides (''p''-1), gcd(''r'', (''p''-1)/r) = 1 and gcd(''q''-1,r) = 1~~.
	*Set ''n'' = ''pq''
	Choose <math>y \in (\mathbb{Z}/n\mathbb{Z})^</math> such that <math>y^{(p-1)(q-1)/r} \not \equiv 1 \mod n</math>.

	The ~~public key is then ''y'',''n'',~~ and ~~the private key is the two primes ''p'',''q''.~~

	~~===Message Encryption===~~
	~~To encrypt a message ''m'', where ''m''~~ is ~~taken~~ to ~~be an element in <math>\mathbb{Z}/r\mathbb{Z}</math>~~

	Choose a random <math>u \in (\mathbb{Z}/n\mathbb{Z})^</math>
	*Set <math>E_r(m) = y^m u^r \mod n</math>

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

	~~To understand decryption, we first notice that for any ''m'',''u'' we have~~

	~~:<math>(y^m u^r)^{(p-1)(q-1)/r} \equiv y^{m(p-1)(q-1)/r} u^{(p-1)(q-1)} \equiv y^{m(p-1)(q-1)/r} \mod n</math>~~

	~~Since ''m'' < ''r'' and <math>y^{(p-1)(q-1)/r} \not \equiv 1 \mod n</math>, we can conclude that <math>(y^m u^r)^{(p-1)(q-1)/r} \equiv 1 \mod n</math> if~~ and ~~only if ''m'' = 0.~~

	~~So if <math>z = y^m u^r \mod n</math> is an encryption of ''m'', given the secret key ''p'',''q'' we can determine whether ''m''=0. If ''r'' is small, we can decrypt ''z''~~ by ~~doing an exhaustive search, i.e~~. ~~decrypting the messages ''y''~~<~~sup~~>~~-''i''~~<~~/sup~~>~~z for ''i'' from 1 to ''r''. By precomputing values, using the [[Baby-step giant-step]] algorithm, decryption can be done in time <math>O~~(~~\sqrt{r})</math>.~~

	~~===Security===~~

	The security of this scheme rests on the [[Higher residuosity problem]], specifically, given ''z'',''r'' and ''n'' where the factorization of ''n'' is unknown, it is computationally infeasible to determine whether ''z'' is an ''r''th residue mod ''n'', i.e. if there exists an ''x'' such that <math>z \equiv x^r \mod n</math>.

	~~==References==~~
	[http://~~research~~.~~microsoft.com~~/~~en-us~~/um/~~people/benaloh~~/~~papers~~/~~dpe~~.~~ps Original Paper~~] ~~(ps~~)
	~~{{Cryptography navbox \| public-key}}~~

	~~[[Category:Public-key encryption schemes]]~~

en>MusikAnimal: Reverted edits by 207.28.222.49 (talk) to last revision by Gilliam (HG)

2014-01-21T16:57:17Z

Reverted edits by 207.28.222.49 (talk) to last revision by Gilliam (HG)

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en>IronGargoyle: Reverted edits by 86.20.107.117 (talk) editing tests or vandalism (HG)

2012-08-26T17:34:10Z

Reverted edits by 86.20.107.117 (talk) editing tests or vandalism (HG)

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Formation and evolution of the Solar System - Revision history

en>JorisvS: /* Galactic interaction */ ce

en>Charles Edwin Shipp: /* See also */ Cosmology

en>MusikAnimal: Reverted edits by 207.28.222.49 (talk) to last revision by Gilliam (HG)

en>IronGargoyle: Reverted edits by 86.20.107.117 (talk) editing tests or vandalism (HG)