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In [[cryptography]] and the [[theory of computation]], the next-bit test<ref name="yao82">[[Andrew Chi-Chih Yao]]. [http://www.busim.ee.boun.edu.tr/~mihcak/teaching/ee684-spring07/proposed-project-papers/one-way-functions/Yao-XOR-Lemma-and-Hard-Core-Predicates/Yao-XOR-original.pdf Theory and applications of trapdoor functions]. In Proceedings of the 23rd IEEE Symposium on Foundations of Computer Science, 1982.</ref> is a test against [[Pseudo-random| pseudo-random number generators]]. We say that a sequence of bits passes the next bit test for at any position <math>i</math> in the sequence, if an attacker knows the <math>i</math> first bits, he cannot predict the <math>(i+1)</math>st with reasonable computational power.
It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.<br><br>Here are some common dental emergencies:<br>Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.<br><br>At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.<br><br>Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.<br><br>Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.<br><br>Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.<br><br>Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.<br><br>Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.<br><br>In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.<br><br>If you want to see more information regarding [http://www.youtube.com/watch?v=90z1mmiwNS8 dentist DC] check out our website.
 
== Precise statement(s) ==
Let <math>P</math> be a polynomial, and <math>S=\{S_k\}</math> be a collection of sets such that <math>S_k</math> contains <math>P(k)</math>-bit long sequences. Moreover, let <math>\mu_k</math> be the [[probability distribution]] of the strings in <math>S_k</math>.
 
We now define the next-bit test in two different ways.
 
===Boolean circuit formulation===
A predicting collection<ref>[[Manuel Blum]] and [[Silvio Micali]], How to generate cryptographically strong sequences of pseudo-random bits, in SIAM J. COMPUT., Vol. 13, No. 4, November 1984</ref> <math>C=\{C_k^i\}</math> is a collection of [[boolean circuits]], such that each circuit <math>C_k^i</math> has less than <math>P_C(k)</math> gates and exactly <math>i</math> inputs. Let <math>p_{k,i}^C</math> be the probability that, on input the <math>i</math> first bits of <math>s</math>, a string randomly selected in <math>S_k</math> with probability <math>\mu_k(s)</math>, the circuit correctly predicts <math>s_{i+1}</math>, i.e. :
<center>
<math>
p_{k,i}^C={\mathcal P} \left[ C_k(s_1\ldots s_i)=s_{i+1} \right | s\in S_k\text{ with probability }\mu_k(s)]
</math>
</center>
Now, we say that <math>\{S_k\}_k</math> passes the next-bit test if for any predicting collection <math>C</math>, any polynomial <math>Q</math> :
<center>
<math>p_{k,i}^C<\frac{1}{2}+\frac{1}{Q(k)}</math>
</center>
 
===Probabilistic Turing machines===
 
We can also define the next-bit test in terms of [[probabilistic Turing machines]], although this definition is somewhat stronger (see [[P/poly#Adleman's theorem|Adleman's theorem]]). Let <math>\mathcal M</math> be a probabilistic Turing machine, working in polynomial time. Let <math>p_{k,i}^{\mathcal M}</math> be the probability that <math>\mathcal M</math> predicts the <math>(i+1)</math>st bit correctly, i.e.
<center>
<math>p_{k,i}^{\mathcal M}={\mathcal P}[M(s_1\ldots s_i)=s_{i+1} | s\in S_k\text{ with probability }\mu_k(s)]</math>
</center>
We say that collection <math>S=\{S_k\}</math> passes the next-bit test if for all polynomial <math>Q</math>, for all but finitely many <math>k</math>, for all <math>0<i<k</math>:  
<center>
<math>
p_{k,i}^{\mathcal M}<\frac{1}{2}+\frac{1}{Q(k)}
</math>
</center>
 
== Completeness for Yao's test ==
 
The next-bit test is a particular case of [[Yao's test]] for random sequences, and passing it is therefore a [[necessary condition]] for passing [[Yao's test]]. However, it has also been shown a [[sufficient condition]] by [[Andrew Chi-Chih Yao|Yao]].<ref name="yao82"/>
 
We prove it now in the case of probabilistic Turing machine, since [[Leonard Adleman|Adleman]] has already done the work of replacing randomization with non-uniformity in [[Adleman's theorem|his theorem]]. The case of boolean circuits cannot be derived from this case (since it involves deciding potentially undecidable problems), but the proof of Adleman's theorem can be easily adapted to the case of non-uniform boolean circuits families.
 
Let <math>\mathcal M</math> a distringuer for the probabilistic version of Yao's test, i.e. a probabilistic Turing machine, running in polynomial time, such that there is a polynomial <math>Q</math> such that for infinitely many <math>k</math>
<center><math>|p_{k,S}^{\mathcal M}-p_{k,U}^{\mathcal M}|\geq\frac{1}{Q(k)}</math></center>
 
Let <math>R_{k,i}=\{s_1\ldots s_iu_{i+1}\ldots u_{P(k)}| s\in S_k, u\in\{0,1\}^{P(k)}\}</math>. We have : <math>R_{k,0}=\{0,1\}^{P(k)}</math> and <math>R_{k,P(k)}=S_k</math>.
Then, we notice that <math>\sum_{i=0}^{P(k)}|p_{k,R_{k,i+1}}^{\mathcal M}-p_{k,R_{k,i}}^{\mathcal M}|\geq |p^{\mathcal M}_{k,R_{k,P(k)}}-p^{\mathcal M}_{k,R_{k,0}}|=|p_{k,S}^{\mathcal M}-p_{k,U}^{\mathcal M}|\geq\frac{1}{Q(k)}</math>. Therefore, at least one of the <math>|p_{k,R_{k,i+1}}^{\mathcal M}-p_{k,R_{k,i}}^{\mathcal M}|</math> should be no smaller than <math>\frac{1}{Q(k)P(k)}</math>.
 
Next, we consider probability distributions <math>\mu_{k,i}</math> and <math>\overline{\mu_{k,i}}</math> on <math>R_{k,i}</math>. Distribution <math>\mu_{k,i}</math> is the probability distribution of choosing the <math>i</math> first bits in <math>S_k</math> with probability given by <math>\mu_k</math>, and the <math>P(k)-i</math> remaining bits uniformly at random. We have thus :
<center>
<math>\mu_{k,i}(w_1\ldots w_{P(k)})=\left(\sum_{s\in S_k, s_1\ldots s_i=w_1\ldots w_i}\mu_k(s)\right)\left(\frac{1}{2}\right)^{P(k)-i}</math>
</center><center>
<math>\overline{\mu_{k,i}}(w_1\ldots w_{P(k)})=\left(\sum_{s\in S_k, s_1\ldots s_{i-1}(1-s_i)=w_1\ldots w_i}\mu_k(s)\right)\left(\frac{1}{2}\right)^{P(k)-i}</math>
</center>
 
We thus have <math>\mu_{k,i}=\frac{1}{2}(\mu_{k,i+1}+\overline{\mu_{k,i+1}})</math> (a simple calculus trick shows this), thus distributions <math>\mu_{k,i+1}</math> and <math>\overline{\mu_{k,i+1}}</math> can be distinguished by <math>\mathcal M</math>. Without loss of generality, we can assume that <math>p^{\mathcal M}_{\mu_{k,i+1}}-p^{\mathcal M}_{\overline{\mu_{k,i+1}}}\geq\frac{1}{2}+\frac{1}{R(k)}</math>, with <math>R</math> a polynomial.
 
This gives us a possible construction of a Turing machine solving the next-bit test : upon receiving the <math>i</math> first bits of a sequence, <math>\mathcal N</math> pads this input with a guess of bit <math>l</math> and then <math>P(k)-i-1</math> random bits, chosen with uniform probability. Then it runs <math>\mathcal M</math>, and outputs <math>l</math> if the result is <math>1</math>, and <math>1-l</math> else.
 
== References ==
<references/>
 
{{DEFAULTSORT:Next-Bit Test}}
[[Category:Pseudorandom number generators]]

Revision as of 16:47, 27 February 2014

It is very common to have a dental emergency -- a fractured tooth, an abscess, or severe pain when chewing. Over-the-counter pain medication is just masking the problem. Seeing an emergency dentist is critical to getting the source of the problem diagnosed and corrected as soon as possible.

Here are some common dental emergencies:
Toothache: The most common dental emergency. This generally means a badly decayed tooth. As the pain affects the tooth's nerve, treatment involves gently removing any debris lodged in the cavity being careful not to poke deep as this will cause severe pain if the nerve is touched. Next rinse vigorously with warm water. Then soak a small piece of cotton in oil of cloves and insert it in the cavity. This will give temporary relief until a dentist can be reached.

At times the pain may have a more obscure location such as decay under an old filling. As this can be only corrected by a dentist there are two things you can do to help the pain. Administer a pain pill (aspirin or some other analgesic) internally or dissolve a tablet in a half glass (4 oz) of warm water holding it in the mouth for several minutes before spitting it out. DO NOT PLACE A WHOLE TABLET OR ANY PART OF IT IN THE TOOTH OR AGAINST THE SOFT GUM TISSUE AS IT WILL RESULT IN A NASTY BURN.

Swollen Jaw: This may be caused by several conditions the most probable being an abscessed tooth. In any case the treatment should be to reduce pain and swelling. An ice pack held on the outside of the jaw, (ten minutes on and ten minutes off) will take care of both. If this does not control the pain, an analgesic tablet can be given every four hours.

Other Oral Injuries: Broken teeth, cut lips, bitten tongue or lips if severe means a trip to a dentist as soon as possible. In the mean time rinse the mouth with warm water and place cold compression the face opposite the injury. If there is a lot of bleeding, apply direct pressure to the bleeding area. If bleeding does not stop get patient to the emergency room of a hospital as stitches may be necessary.

Prolonged Bleeding Following Extraction: Place a gauze pad or better still a moistened tea bag over the socket and have the patient bite down gently on it for 30 to 45 minutes. The tannic acid in the tea seeps into the tissues and often helps stop the bleeding. If bleeding continues after two hours, call the dentist or take patient to the emergency room of the nearest hospital.

Broken Jaw: If you suspect the patient's jaw is broken, bring the upper and lower teeth together. Put a necktie, handkerchief or towel under the chin, tying it over the head to immobilize the jaw until you can get the patient to a dentist or the emergency room of a hospital.

Painful Erupting Tooth: In young children teething pain can come from a loose baby tooth or from an erupting permanent tooth. Some relief can be given by crushing a little ice and wrapping it in gauze or a clean piece of cloth and putting it directly on the tooth or gum tissue where it hurts. The numbing effect of the cold, along with an appropriate dose of aspirin, usually provides temporary relief.

In young adults, an erupting 3rd molar (Wisdom tooth), especially if it is impacted, can cause the jaw to swell and be quite painful. Often the gum around the tooth will show signs of infection. Temporary relief can be had by giving aspirin or some other painkiller and by dissolving an aspirin in half a glass of warm water and holding this solution in the mouth over the sore gum. AGAIN DO NOT PLACE A TABLET DIRECTLY OVER THE GUM OR CHEEK OR USE THE ASPIRIN SOLUTION ANY STRONGER THAN RECOMMENDED TO PREVENT BURNING THE TISSUE. The swelling of the jaw can be reduced by using an ice pack on the outside of the face at intervals of ten minutes on and ten minutes off.

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