Schmoyer, Susan LynnLet E be an elliptic curve defined over F_q and suppose that E[n] subset E(F_q). For attacking the elliptic curve discrete logarithm problem it is useful to know when points pair with themselves nontrivially under the Tate-Lichtenbaum pairing. In this thesis we characterize when all points in E[n] have trivial self pairings. This result is expressed in terms of the action of the Frobenius endomorphism on E[n^2]. We then generalize this result to Jacobians of algebraic curves of arbitrary genus.en-USDissertationMathematicsMathematics