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SafeCurves lists some ways to compare the security of elliptic curves. Their security criteria are split to "ECDLP security" and "ECC security". Failing the former basically means "there is no way to use this curve securely in general" while the latter "it is difficult to implement this curve securely". None of the (few) BouncyCastle-supported curves that ...

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You may probably use any curve you like, depending on your special requirements (environment, computational aspects, ...) and the curves implemented by your library (see otus answer refering to some concrete security findings related to specific elliptic curves, and how sensible they are to certain attacks). The reason why the curves are pre-computed, is ...

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Choose some 128-bit hash function, such as RIPEMD-128, and a way of randomizing it, such as this. The private key is either 60 uniformly random 128-bit strings s00,s01,...,s58,s59 and a uniformly random short salt or a seed to regenerate those. $\:$ For each i in {00,01,...,58,59}, vi is the result of hashing si 19 times. $\:$ The public key is the ...

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You can use multi-signatures. One example is the BN06 scheme described in the paper: Bellare, Neven - Multi-signatures in the plain public-Key model and a general forking lemma

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I don't think a reduction based on repeated messages with possible with normal ECC signatures (ECDSA, ElGamal, EdDSA, etc.). A few alternative approaches: public key recovery This doesn't reduce the size of the signature itself, but you don't need to transmit the public key alongside the signature. If applicable, this gains you 2 times the security level ...

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This may be way too late, but you could have a look at CryptoNote. It's designed for cryptocurrencys but should be suitable for your purposes. I think there is an implementation somewhere in the Bytecoin code.

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Informally, a signature scheme with message recovery is one where some or all of the message is embedded in the signature, allowing to conserve bandwidth when transmitting a signed message, compared to a signature scheme with appendix. Total message recovery A signature scheme with total message recovery [some sources make total implicit, e.g. the HAC ...

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