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message, to single wolves they indicate a mated pair, and to other pairs they warn against territorial intrusion (Peters and Mech 1975b; Rothman and Mech 1979). 86 Fred H. Harrington and Cheryl S. Asa A particular feature of female wolf reproductive phys- iology associated with urine marking that may facili- tate pair bonding is the relatively long proestrous period (average 6 weeks: Asa et al. 1986) compared with that of the dog (1 week: Christie and Bell 1971a). Proestrus is preceded by a transient increase in testosterone and ac- companied by elevated estrogen

For which groups (of the same prime order p) used in cryp- tographic protocols and which values i, 1 ≤ i ≤ p− 1, do efficient algorithms for computing ei exist? More generally, G can be E[m]; for the Tate pairing, efficient algorithms with performance comparable to that of RSA have been found [21]. 9.1.3 Cocyclic codes Many good error-correcting block codes (see Chapter 3.2.1) are derived from v× v matrices M with entries in a commutative ring R with unity, which have in addition some internal structure. The rows themselves may form the code. For example, the rows

holomorphic projection of the linear combination of products of some modular forms. We call Z the geometric kernel and call Φ the analytic kernel. Then the geometric kernel Z gives the Néron-Tate height, and the analytic kernel Φ gives the L-function. The identity (4.60) in the proof of the original Gross-Zagier formula is a particular case of (4.61). Proof of the fundamental identity. Let us give a brief explanation on how to prove the equality (4.61). The definition of Z involves the Néron-Tate pairing and Hecke correspondence. Hence the terms of Z can be regrouped into