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The security of elliptic and hyperelliptic curve cryptosystems is based on the computational difficulty of solving the Discrete Logarithm Problem (DLP). There is currently no sub-exponential algorithm for solving the discrete logarithm problem on the Jacobians of properly chosen curves. In cryptographic applications, hyperelliptic curves of small genus have the advantage of providing a group of comparable size to that of elliptic curves, while working over a field of smaller size. Pairing-friendly hyperelliptic curves are those for which the order of the Jacobian is divisible by a large prime, whose embedding degree is small enough for pairing computations to be feasible, and whose minimal embedding field is large enough for the discrete logarithm problem in it to be difficult.
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