Power of power modulo with non square-free numbers

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Reference https://www.hackerrank.com/challenges/devu-police/problem



I have five numbers a, b, c, d, N and I need to find $(a^b)^c^d$ mod N. N is non square-free number so euler totient theorem or CRT won't work here.



I have tried this way but according to test-cases my program gets timed out. I find $a^b$ mod N then I run power modulo with algorithm https://www.geeksforgeeks.org/modular-exponentiation-power-in-modular-arithmetic/ here exponent divided by 2 and I do with c so each computation needs O(d) which should not be rather logarithm solution expected.



Could you please help me to solve?




number-theory




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  • Euler's totient theorem does not need a squarefree modulo, it is enough that the base is coprime to the modulo.
    – Peter
    Jul 28 at 11:45










  • This will be useful for your problem : en.wikipedia.org/wiki/Carmichael_function
    – Peter
    Jul 28 at 11:51










  • @Peter if they r not co-prime then what?
    – Debasis Jana
    Jul 28 at 12:08










  • If you know the factorization of $N$, my link should solve the problem.
    – Peter
    Jul 28 at 12:19










  • @Peter thank you ... :) actually I didn't go through your link .. ;)
    – Debasis Jana
    Jul 28 at 12:36














up vote
0
down vote

favorite












Reference https://www.hackerrank.com/challenges/devu-police/problem



I have five numbers a, b, c, d, N and I need to find $(a^b)^c^d$ mod N. N is non square-free number so euler totient theorem or CRT won't work here.



I have tried this way but according to test-cases my program gets timed out. I find $a^b$ mod N then I run power modulo with algorithm https://www.geeksforgeeks.org/modular-exponentiation-power-in-modular-arithmetic/ here exponent divided by 2 and I do with c so each computation needs O(d) which should not be rather logarithm solution expected.



Could you please help me to solve?




number-theory




share|cite|improve this question





















  • Euler's totient theorem does not need a squarefree modulo, it is enough that the base is coprime to the modulo.
    – Peter
    Jul 28 at 11:45










  • This will be useful for your problem : en.wikipedia.org/wiki/Carmichael_function
    – Peter
    Jul 28 at 11:51










  • @Peter if they r not co-prime then what?
    – Debasis Jana
    Jul 28 at 12:08










  • If you know the factorization of $N$, my link should solve the problem.
    – Peter
    Jul 28 at 12:19










  • @Peter thank you ... :) actually I didn't go through your link .. ;)
    – Debasis Jana
    Jul 28 at 12:36












up vote
0
down vote

favorite









up vote
0
down vote

favorite











Reference https://www.hackerrank.com/challenges/devu-police/problem



I have five numbers a, b, c, d, N and I need to find $(a^b)^c^d$ mod N. N is non square-free number so euler totient theorem or CRT won't work here.



I have tried this way but according to test-cases my program gets timed out. I find $a^b$ mod N then I run power modulo with algorithm https://www.geeksforgeeks.org/modular-exponentiation-power-in-modular-arithmetic/ here exponent divided by 2 and I do with c so each computation needs O(d) which should not be rather logarithm solution expected.



Could you please help me to solve?




number-theory




share|cite|improve this question













Reference https://www.hackerrank.com/challenges/devu-police/problem



I have five numbers a, b, c, d, N and I need to find $(a^b)^c^d$ mod N. N is non square-free number so euler totient theorem or CRT won't work here.



I have tried this way but according to test-cases my program gets timed out. I find $a^b$ mod N then I run power modulo with algorithm https://www.geeksforgeeks.org/modular-exponentiation-power-in-modular-arithmetic/ here exponent divided by 2 and I do with c so each computation needs O(d) which should not be rather logarithm solution expected.



Could you please help me to solve?





number-theory





share|cite|improve this question












share|cite|improve this question




share|cite|improve this question








edited Jul 28 at 8:14
























asked Jul 28 at 7:47









Debasis Jana

63




63











  • Euler's totient theorem does not need a squarefree modulo, it is enough that the base is coprime to the modulo.
    – Peter
    Jul 28 at 11:45










  • This will be useful for your problem : en.wikipedia.org/wiki/Carmichael_function
    – Peter
    Jul 28 at 11:51










  • @Peter if they r not co-prime then what?
    – Debasis Jana
    Jul 28 at 12:08










  • If you know the factorization of $N$, my link should solve the problem.
    – Peter
    Jul 28 at 12:19










  • @Peter thank you ... :) actually I didn't go through your link .. ;)
    – Debasis Jana
    Jul 28 at 12:36
















  • Euler's totient theorem does not need a squarefree modulo, it is enough that the base is coprime to the modulo.
    – Peter
    Jul 28 at 11:45










  • This will be useful for your problem : en.wikipedia.org/wiki/Carmichael_function
    – Peter
    Jul 28 at 11:51










  • @Peter if they r not co-prime then what?
    – Debasis Jana
    Jul 28 at 12:08










  • If you know the factorization of $N$, my link should solve the problem.
    – Peter
    Jul 28 at 12:19










  • @Peter thank you ... :) actually I didn't go through your link .. ;)
    – Debasis Jana
    Jul 28 at 12:36















Euler's totient theorem does not need a squarefree modulo, it is enough that the base is coprime to the modulo.
– Peter
Jul 28 at 11:45




Euler's totient theorem does not need a squarefree modulo, it is enough that the base is coprime to the modulo.
– Peter
Jul 28 at 11:45












This will be useful for your problem : en.wikipedia.org/wiki/Carmichael_function
– Peter
Jul 28 at 11:51




This will be useful for your problem : en.wikipedia.org/wiki/Carmichael_function
– Peter
Jul 28 at 11:51












@Peter if they r not co-prime then what?
– Debasis Jana
Jul 28 at 12:08




@Peter if they r not co-prime then what?
– Debasis Jana
Jul 28 at 12:08












If you know the factorization of $N$, my link should solve the problem.
– Peter
Jul 28 at 12:19




If you know the factorization of $N$, my link should solve the problem.
– Peter
Jul 28 at 12:19












@Peter thank you ... :) actually I didn't go through your link .. ;)
– Debasis Jana
Jul 28 at 12:36




@Peter thank you ... :) actually I didn't go through your link .. ;)
– Debasis Jana
Jul 28 at 12:36















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