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Math Help - Solving modular inequality

  1. #1
    Super Member Bacterius's Avatar
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    Solving modular inequality

    Hello,
    I'm trying to solve the following inequality efficiently. I am given a composite number n with unknown factorization (it shall not be attempted to factor it), and a composite number k with known factorization. What is more, \varphi{(k)} is known along with its factorization.

    A solution is a c such that :

    c^{\varphi{(k)}} \equiv a \pmod{kn}

    And a < \sqrt[\left \lceil \ln{k} \right \rceil]{kn}.

    This seems impossible somehow.
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  2. #2
    MHF Contributor chiph588@'s Avatar
    Joined
    Sep 2008
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    Champaign, Illinois
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    1,163
    Quote Originally Posted by Bacterius View Post
    Hello,
    I'm trying to solve the following inequality efficiently. I am given a composite number n with unknown factorization (it shall not be attempted to factor it), and a composite number k with known factorization. What is more, \varphi{(k)} is known along with its factorization.

    A solution is a c such that :

    c^{\varphi{(k)}} \equiv a \pmod{kn}

    And a < \sqrt[\left \lceil \ln{k} \right \rceil]{kn}.

    This seems impossible somehow.
    I assume  a\geq0 ?
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