Dual basis in a field extension: Difference between revisions

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The '''quadratic residuosity problem''' in [[computational number theory]] is the question of distinguishing by calculating the [[quadratic residue]]s [[modular arithmetic|modulo]] ''N'', where ''N'' is a [[composite number]]. This is an important consideration in contemporary [[cryptography]].<ref>Mathematicians say ''residuacity'': ''residuosity'', something of a [[malapropism]], has been adopted by most cryptographers.</ref>
 
==Formulation==
Given the specific case of ''N'' being the product of distinct odd [[prime number]]s ''p'' and ''q'', the structure of the squaring map:
 
:''a'' &rarr; ''a''<sup>2</sup> mod ''N''
 
on the [[Multiplicative group of integers modulo n|multiplicative group of invertible residues modulo ''N'']], is as a [[group homomorphism]] with [[kernel (algebra)|kernel]] a [[Klein group]] of order four. The image is therefore of size roughly ''N''/4.  More precisely, it is of order:
 
:<math>\frac{(p - 1)(q - 1)}{4}</math>
 
In contrast, the same mapping modulo prime ''P'' has the kernel of order 2 and the image of order (''P'' &minus; 1)/2. In this case it is easy to characterize the image computationally, since the [[Jacobi symbol]] takes the value +1 precisely on quadratic residues modulo ''P''.
 
Modulo composite ''N'' the corresponding Jacobi symbol characterizes a [[subgroup]] of the residues which is larger by factor of two; that is, it rules out roughly half of the residues modulo ''N'', while the problem as posed is to characterize a subset of size a quarter of ''N''. This difference constitutes the quadratic residuosity problem, in this particular but essential case of ''N'' being the product of two primes.
 
The [[computational hardness assumption]] is that bridging this gap can only to be done by lengthy calculation, when quantified in terms of the size of ''N''.
 
==Applications==
The intractability of the quadratic residuosity problem is the basis for the security of the [[Blum Blum Shub]] pseudo-random number generator and the [[Goldwasser–Micali cryptosystem]].<ref>{{cite journal |author=S. Goldwasser, S. Micali |title=Probabilistic encryption and how to play mental poker keeping secret all partial information |journal=Proc. 14th Symposium on Theory of Computing |year=1982 |pages=365–377 |doi=10.1145/800070.802212}}</ref><ref>{{cite journal |author=S. Goldwasser, S. Micali |title=Probabilistic encryption |journal=Journal of Computer and System Sciences |volume=28 |issue=2 |year=1984 |pages=270–299 |doi=10.1016/0022-0000(84)90070-9}}</ref>
 
==See also==
* [[Higher residuosity problem]]
 
==Notes==
{{Reflist}}
 
 
[[Category:Computational number theory]]
[[Category:Computational hardness assumptions]]
[[Category:Theory of cryptography]]

Latest revision as of 06:58, 25 April 2014

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