Let A be an abelian variety defined over a non-prime finite field F_q that has embedding degree k with respect to a subgroup of prime order r. In this paper we give explicit conditions on q, k, and r that imply that the minimal embedding field of A with respect to r is F_(q~k). When these conditions hold, the embedding degree k is a good measure of the security level of a pairing-based cryptosystem that uses A. We apply our theorem to supersingular elliptic curves and to super-singular genus 2 curves, in each case computing a maximum ρ-value for which the minimal embedding field must be F_(q~k) . Our results are in most cases stronger (i.e., give larger allowable ρ-values) than previously known results for supersingular varieties, and our theorem holds for general abelian varieties, not only supersingular ones.
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机译：让A在非Prime Unitipe Field F_Q上定义的Abelian品种，该字段F_Q具有嵌入程度k的基本组r。在本文中，我们在Q，K和R上给出了明确条件，这意味着关于R的最小嵌入字段是F_（Q〜K）。当这些条件保持时，嵌入程度k是使用A的配对基密码系统的安全级别的良好衡量标准。我们将定理应用于超出椭圆曲线和超奇异的属2曲线，在每种情况下计算最大最小嵌入字段必须为f_（q〜k）的ρ值。我们的结果在大多数情况下更强（即，给予较大的允许ρ值）而不是先前已知的超周定品种的结果，以及我们的定理为普通的阿贝尔品种，不仅是超出的。
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