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| In 1997, [[Moni Naor]] and [[Omer Reingold]] described efficient constructions for various [[cryptographic primitive]]s in private key as well as [[public-key cryptography]]. Their result is the construction of an efficient [[pseudorandom function family|pseudorandom function]]. Let ''p'' and ''l'' be [[prime number]]s with ''l'' |''p''-1. Select an element ''g'' ∈ <math> {\mathbb F_p}^*</math> of [[multiplicative order]] ''l''. Then for each n-dimensional [[coordinate vector|vector]] ''a'' = (''a''<sub>''1''</sub>, ..., ''a''<sub>''n''</sub>)∈ <math> (\mathbb F_{l})^{n} </math> they define the function
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| :<math>f_{a}(x) = g^{a_{1}^{x_{1}} a_{2}^{x_{2}}...a_{n}^{x_{n}}} \in \mathbb F_p </math>
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| where x = x<sub>''1''</sub> ... x<sub>''n''</sub> is the [[Binary_numeral_system#Representation|bit representation]] of integer x, 0 ≤ x ≤ 2<sup>n-1</sup>, with some extra leading zeros if necessary.<ref name="NaorReingold"/>
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| ==Example==
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| Let ''p'' = 7, ''p'' – 1 = 6, and ''l'' = 3, ''l'' |''p''-1. Select ''g'' = 4 ∈ <math> {\mathbb F_7}^*</math> of multiplicative order 3 (since 4<sup>3</sup> = 64 ≡ 1 mod 7). For n = 3, a = (1, 2, 1) and x = 5 (the bit representation of 5 is 101), we can compute <math> f_{a}\;(5)</math> as follows:
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| :<math>f_{a}(x) = g^{a_{1}^{x_{1}} a_{2}^{x_{2}}...a_{n}^{x_{n}}}</math>
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| :<math>f_{a}(5) = 4^{1^{1} 2^{0} 1^{1}} = 4^{1} = 4 \in \mathbb F_7 </math>
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| ==Efficiency==
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| The evaluation of function <math> f_{a}\;(x)</math> in the '''Naor-Reingold''' construction can be done very efficiently. Computing the value of the function <math> f_{a}\;(x)</math> at any given point is comparable with one [[modular exponentiation]] and n-modular multiplications. This function can be computed in parallel by threshold circuits of bounded depth and polynomial size.
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| The '''Naor-Reingold''' function can be used as the basis of many [[cryptography|cryptographic]] schemes including [[Symmetric-key algorithm|symmetric encryption]], [[authentication]] and [[digital signatures]]. | |
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| ==Security of the Function==
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| Assume that an attacker sees several outputs of the function, e.g. <math> f_{a}(1) = g^{a_{1}}, f_{a}(2) = g^{a_{2}}, f_{a}(3) = g^{a_{1}a_{2}}</math>, ... <math>f_{a}(k) = g^{a_{1}^{x_{1}} a_{2}^{x_{2}}...a_{n}^{x_{n}}}</math> and wants to compute <math> f_{a}\;(k + 1)</math>. Assume for simplicity that x<sub>1</sub> = 0, then the attacker needs to solve the [[Computational Diffie–Hellman assumption|Computational Diffie-Hellman (CDH)]] between <math> f_a (1)= g^{a_{1}} </math> and <math>f_{a}(k) = g^{a_{2}^{x_{2}} ...a_{n}^{x_{n}}}</math> to get <math>f_{a}(k+1) = g^{a_{1}a_{2}^{x_{2}} \dots a_{n}^{x_{n}}}</math>. In general, moving from ''k'' to ''k'' +1 changes the bit pattern and unless ''k'' + 1 is a power of 2 one can split the exponent in <math> f_{a}\;(k + 1)</math> so that the computation corresponds to computing the [[Diffie-Hellman problem|Diffie-Hellman]] key between two of the earlier results. This attacker wants to predict the next [[sequence]] element. Such an attack would be very bad—but it's also possible to fight it off by working in [[Group (mathematics)|groups]] with a hard [[Diffie-Hellman problem]] (DHP).
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| '''Example:'''
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| An attacker sees several outputs of the function e.g. <math>f_{a}(5) = 4^{1^{1} 2^{0} 1^{1}} = 4^{1} = 4 </math>, as in the previous example, and <math>f_{a}(1) = 4^{1^{0} 2^{0} 1^{1}} = 4^{1} = 4 </math>. Then, the attacker wants to predict the next sequence element of this function, <math> f_{a}\;(6)</math>. However, the attacker cannot predict the outcome of <math> f_{a}\;(6) </math> from knowing <math> f_{a}\;(1)</math> and <math> f_{a}\;(5)</math>.
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| There are other attacks that would be very bad for a [[Pseudorandom Number Generator]]: the user expects to get random numbers from the output, so of course the stream should not be predictable, but even more, it should be indistinguishable from a random string. Let <math> \mathcal{A}^f </math> denote the algorithm <math> \mathcal{A} </math> with access to an oracle for evaluating the function <math> f_{a}\;(x)</math> . Suppose the [[Decisional Diffie-Hellman assumption]] holds for <math> \mathbb F_p </math>, '''Naor and Reingold''' show that for every [[PP (complexity)|probabilistic polynomial time]] algorithm <math> \mathcal{A} </math> and sufficiently large ''n''
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| : <math> \text{Pr }[\mathcal{A}^{f_{a}(x)}(p,g) \to 1] - \text{Pr }[\mathcal{A}^{R} (p,g)\to 1] </math> is [[Negligible function|negligible]].
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| The first probability is taken over the choice of the seed s = (p, g, a) and the second probability is taken over the random distribution induced on p, g by <math> \mathcal{I}\mathcal{G} (n) </math>, instance generator, and the random choice of the function <math> R_{a}\;(x) </math> among the set of all <math> \{0,1\}^{n} \to \mathbb F_p </math> functions.<ref name="DDHBoneh"/>
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| ==Linear Complexity==
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| One natural measure of how useful a sequence may be for [[cryptography|cryptographic]] purposes is the size of its '''linear complexity'''. The linear complexity of an ''n''-element sequence W(x), x = 0,1,2,…,n – 1, over a ring <math> \mathcal{R}</math> is the length ''l'' of the shortest linear [[recurrence relation]] W (x + ''l'') = A<sub>''l''-1</sub> W (x +''l''-1) + … + A<sub>0</sub> W(x), x = 0,1,2,…, ''n'' – ''l'' –1 with A<sub>0</sub>, …, A<sub>''l''-1</sub> ∈ <math> \mathcal{R}</math>, which is satisfied by this sequence.
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| For some <math>\gamma</math> > 0,n ≥ (1+ <math>\gamma</math>) <math>\log l</math>, for any <math>\delta > 0 </math>, sufficiently large ''l'', the linear complexity of the sequence <math> f_{a}\;(x)</math>,0 ≤ x ≤ 2<sup>n-1</sup>, denoted by <math>L_a</math> satisfies
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| :<math>L_{a} \geqslant \begin{cases}
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| l^{1-\ \delta\,\!} &\text{, if } \gamma\,\! \geqslant 2\\
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| l^{\left (\tfrac{\ \gamma\,\!}{2-\ \delta\,\!}\right )} &\text{, if } \gamma\,\! < 2
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| \end{cases}</math>
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| for all except possibly at most <math>3(l - 1)^{n - \delta}</math> vectors a ∈ <math> (\mathbb F_{l})^{n} </math>.<ref name="ShparlinskiLinearComplexity"/> The bound of this work has disadvantages, namely it does not apply to the very interesting case <math>\log p \approx \log n \approx {n.} </math>
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| ==Uniformity of Distribution==
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| The statistical distribution of <math> f_{a}\;(x)</math> is exponentially close to [[Uniform distribution (discrete)|uniform distribution]] for almost all vectors a ∈ <math> (\mathbb F_{l})^{n} </math>.
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| Let <math>{\mathbf D}_a</math> be the [[Inversive congruential generator|discrepancy]] of the set <math>\{f_a (x)| 0 \leq x \leq 2^{n-1}\}</math>. Thus, if <math>n = \log p </math> is the bit length of ''p'' then for all vectors a ∈ <math> (\mathbb F_{l})^{n} </math> the bound <math>{\mathbf D}_a\leq \Delta (l,p)</math> holds, where
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| <math>\Delta (l,p) = \begin{cases}
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| p^{\left (\tfrac{1-\ \gamma\,\!}{2}\right )}l^{\left (\tfrac{-1}{2}\right )}\log^{2}p &\text{, if } l \geqslant p^{\gamma\,\!}\\
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| p^{\left (\tfrac{1}{2}\right )}l^{-1}\log^{2}p &\text{, if } p^{\gamma\,\!} > l \geqslant p^{\left (\tfrac{2}{3}\right )} \\
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| p^{\left (\tfrac{1}{4}\right )}l^{\left (\tfrac{-5}{8}\right )}\log^{2}p &\text{, if } p^{\left (\tfrac{2}{3}\right )} > l \geqslant p^{\left (\tfrac{1}{2}\right )} \\
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| p^{\left (\tfrac{1}{8}\right )}l^{\left (\tfrac{-3}{8}\right )}\log^{2}p &\text{, if } p^{\left (\tfrac{1}{2}\right )} > l \geqslant p^{\left (\tfrac{1}{3}\right )} \\
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| \end{cases}</math>
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| and <math>\gamma</math> = 2.5 - <math>\log 3 </math>= 0.9150....
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| Although this property does not seem to have any immediate cryptographic implications, the inverse fact, namely non uniform distribution, if true would have disastrous consequences for applications of this function.<ref name="ShparlinskiUniformity"/>
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| ==Sequences in Elliptic Curve==
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| The [[elliptic curve]] version of this function is of interest as well. In particular, it may help to improve the cryptographic security of the corresponding system. Let ''p'' > 3 be prime and let E be an elliptic curve over <math> \mathbb F_p </math>, then each vector '''a''' defines a [[Sequence#Finite_and_infinite|finite sequence]] in the [[subgroup]] <math>\langle G\rangle</math> as:
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| :<math>F_{a}(x) = (a_{1}^{x_{1}} a_{2}^{x_{2}}\dots a_{n}^{x_{n}})G </math>
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| where <math>x = x_1 \dots x_n</math> is the bit representation of integer <math>x, 0 \leq x \leq 2^{n-1}</math>.
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| The '''Naor-Reingold''' elliptic curve sequence is defined as
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| : <math> u_{k} = X (f_{a}(k))\; \mbox{where } X (P) \mbox{ is the abscissa of}\; P \in E. </math><ref name="EllipticNaor"/>
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| If the Decisional Diffie-Hellman assumption holds, the index ''k'' is not enough to compute <math>u_k</math> in polynomial time, even if an attacker performs polynomially many queries to a random oracle.
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| ==See also==
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| *[[Decisional Diffie-Hellman assumption]]
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| *[[Finite Field]]
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| *[[Inversive congruential generator]]
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| *[[Generalized inversive congruential pseudorandom numbers]]
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| ==Notes==
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| {{Reflist|refs=
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| <ref name="NaorReingold">Naor, M., Reingold, O. "Number-theoretic constructions of efficient pseudo-random functions," Proc 38th IEEE Symp. on Foundations of Comp. Sci, (1997), 458-467.</ref>
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| <ref name="ShparlinskiLinearComplexity">Shparlinski, Igor E. "Linear Complexity of the Naor-Reingold pseudo-random function," Inform. Process Lett, 76 (2000), 95-99.</ref>
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| <ref name="ShparlinskiUniformity">Shparlinski, Igor E. "On the uniformity of distribution of the Naor-Reingold pseudo-random function," Finite Fields and Their Applications, 7 (2001), 318-326</ref>
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| <ref name="EllipticNaor">Cruz, M., Gomez, D., Sadornil, D. "On the linear complexity of the Naor-Reingold sequence with elliptic curves," Finite Fields and Their Applications, 16 (2010), 329-333</ref>
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| <ref name="DDHBoneh">Boneh, Dan. "The Decision Diffie–Hellman Problem,"ANTS-III: Proceedings of the Third International Symposium on Algorithmic Number Theory,1998,48–63.</ref>
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| }}
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| ==References==
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| *{{citation
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| | last=Shparlinski
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| | first=Igor
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| | title=Cryptographic Applications of Analytic Number Theory: Complexity Lower Bounds and Pseudorandomness
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| | year=2003
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| | edition=first
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| | publisher=Birkhäuser Basel
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| | isbn=978-3-7643-6654-4
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| }}
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| *{{citation
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| | last=Goldreich
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| | first=Oded
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| | title=Modern Cryptography, Probabilistic Proofs and Pseudorandomness
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| | year=1998
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| | edition=first
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| | publisher=Springer
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| | isbn=978-3-540-64766-9
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| }}
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| [[Category:Pseudorandom number generators]]
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| [[Category:Cryptography]]
| |
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Exactly where other types of develop lights need a lot of space and will frequently need a lot of equipment to run, LED grow lights cost-free up your space. The somewhat compact grow lamps will make it achievable to move far more freely all through the develop area, since you will not have to be concerned about added equipment and cooling systems that commonly take up so a lot of the accessible area. The temperature of your grow area is extremely significant. LED grow lights run extremely cool compared to other possibilities.
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