Is exponentiation in P?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











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I think the following problem belong to class P, but I don't know how I can prove it, could somebody help me?



  • Inputs: two numbers $(a,b) in mathbbN$

  • Output: $a^b$





complexity-theory complexity-classes







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  • I don't really see the connection between your two questions, so I have removed the second one.
    – Yuval Filmus
    4 hours ago















up vote
3
down vote

favorite












I think the following problem belong to class P, but I don't know how I can prove it, could somebody help me?



  • Inputs: two numbers $(a,b) in mathbbN$

  • Output: $a^b$





complexity-theory complexity-classes







share|cite|improve this question























  • I don't really see the connection between your two questions, so I have removed the second one.
    – Yuval Filmus
    4 hours ago













up vote
3
down vote

favorite









up vote
3
down vote

favorite











I think the following problem belong to class P, but I don't know how I can prove it, could somebody help me?



  • Inputs: two numbers $(a,b) in mathbbN$

  • Output: $a^b$





complexity-theory complexity-classes







share|cite|improve this question















I think the following problem belong to class P, but I don't know how I can prove it, could somebody help me?



  • Inputs: two numbers $(a,b) in mathbbN$

  • Output: $a^b$





complexity-theory complexity-classes




complexity-theory complexity-classes






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share|cite|improve this question













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edited 4 hours ago









Yuval Filmus

184k12175335




184k12175335










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linkho

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  • I don't really see the connection between your two questions, so I have removed the second one.
    – Yuval Filmus
    4 hours ago

















  • I don't really see the connection between your two questions, so I have removed the second one.
    – Yuval Filmus
    4 hours ago
















I don't really see the connection between your two questions, so I have removed the second one.
– Yuval Filmus
4 hours ago





I don't really see the connection between your two questions, so I have removed the second one.
– Yuval Filmus
4 hours ago











1 Answer
1






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3
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Your problem is not in P, for two different reasons:



  1. P is a class of decision problems, but your problem is a function problem. Instead of P, you should consider its functional equivalent FP.


  2. The output could be exponentially large in the input length: encoding $b$ takes about $log b$ bits, but encoding $a^b$ takes about $b log a$ bits.


This still leaves open the possibility that the following problem is in P:




Given natural numbers $a,b$ and an index $i$, determine the $i$th bit of $a^b$.




While I don't know what the answer to this question is, here is a related problem in FP:




Given natural numbers $a,b,c$, determine $a^b bmod c$.




This can be shown using the important technique of repeated squaring.






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    1 Answer
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    1 Answer
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    up vote
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    down vote













    Your problem is not in P, for two different reasons:



    1. P is a class of decision problems, but your problem is a function problem. Instead of P, you should consider its functional equivalent FP.


    2. The output could be exponentially large in the input length: encoding $b$ takes about $log b$ bits, but encoding $a^b$ takes about $b log a$ bits.


    This still leaves open the possibility that the following problem is in P:




    Given natural numbers $a,b$ and an index $i$, determine the $i$th bit of $a^b$.




    While I don't know what the answer to this question is, here is a related problem in FP:




    Given natural numbers $a,b,c$, determine $a^b bmod c$.




    This can be shown using the important technique of repeated squaring.






    share|cite|improve this answer
























      up vote
      3
      down vote













      Your problem is not in P, for two different reasons:



      1. P is a class of decision problems, but your problem is a function problem. Instead of P, you should consider its functional equivalent FP.


      2. The output could be exponentially large in the input length: encoding $b$ takes about $log b$ bits, but encoding $a^b$ takes about $b log a$ bits.


      This still leaves open the possibility that the following problem is in P:




      Given natural numbers $a,b$ and an index $i$, determine the $i$th bit of $a^b$.




      While I don't know what the answer to this question is, here is a related problem in FP:




      Given natural numbers $a,b,c$, determine $a^b bmod c$.




      This can be shown using the important technique of repeated squaring.






      share|cite|improve this answer






















        up vote
        3
        down vote










        up vote
        3
        down vote









        Your problem is not in P, for two different reasons:



        1. P is a class of decision problems, but your problem is a function problem. Instead of P, you should consider its functional equivalent FP.


        2. The output could be exponentially large in the input length: encoding $b$ takes about $log b$ bits, but encoding $a^b$ takes about $b log a$ bits.


        This still leaves open the possibility that the following problem is in P:




        Given natural numbers $a,b$ and an index $i$, determine the $i$th bit of $a^b$.




        While I don't know what the answer to this question is, here is a related problem in FP:




        Given natural numbers $a,b,c$, determine $a^b bmod c$.




        This can be shown using the important technique of repeated squaring.






        share|cite|improve this answer












        Your problem is not in P, for two different reasons:



        1. P is a class of decision problems, but your problem is a function problem. Instead of P, you should consider its functional equivalent FP.


        2. The output could be exponentially large in the input length: encoding $b$ takes about $log b$ bits, but encoding $a^b$ takes about $b log a$ bits.


        This still leaves open the possibility that the following problem is in P:




        Given natural numbers $a,b$ and an index $i$, determine the $i$th bit of $a^b$.




        While I don't know what the answer to this question is, here is a related problem in FP:




        Given natural numbers $a,b,c$, determine $a^b bmod c$.




        This can be shown using the important technique of repeated squaring.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 3 hours ago









        Yuval Filmus

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