|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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.
| | 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] |
| | |
| ==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]]
| |
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.
My web site - email psychic readings