Algebraic geometers and cryptographers are very familiar with what we like to call the "imaginary model" of a hyperelliptic curve. Another less familiar description of such a curve is the so-called "real model"; the terminology stems from the analogy to real and imaginary quadratic number fields. Structurally and arithmetically, the real model behaves quite differently from its imaginary counterpart. While divisor addition with subsequent reduction ("giant steps") is still essentially the same, the real modedl no longer allows for efficiently computable unique representation of elements in the Jacobian via reduced representatives.
However, the real model exhibits a so-called infrastructure, with an additional much faster operation ("baby steps"). We present the real model of a hyperelliptic curve and its two-fold baby step giant step divisor arithmetic. We also indicate how to use these algorithms for potential cryptographic and number theoretic applications.
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