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I have read that in cases where there is a limit on the amount of data sent over the network, one idea is to sent only $t$ bits of data rather than sending all the bits. For example, in the Schnorr scheme this can be employed.

My question is, whether this can be specified by sending the $t$ bits along with the other information?

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If you want shorter signatures you should consider BLS –  CodesInChaos Jan 29 at 11:58
    
@CodesInChaos Is it possible to send only a part of signature? Like 16 bits? –  user5507 Jan 31 at 2:43
    
There are short MACs (but even they need 64 bits for a decent level of securiy), but even BLS signatures need 2x the security level. One can use proof-of-work to shave of a 20 bits or so. –  CodesInChaos Jan 31 at 8:35
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