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-
tatepair 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 Tatepairing, efficient algorithms with
performance comparable to that of RSA have been found .
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-Tatepairing
and Hecke correspondence. Hence the terms of Z can be regrouped into