To pay for items of arbitrary values, we may need to combine several
tokens in a single transaction. In this case 2 must contain
the various tokens, and be signed with all keys associated with those
tokens. Let 
be a list of tokens (possibly of
different brands) 
and
let 
be the list of corresponding private halves
. We extend the subscript notation to vectors to
indicate signing the plaintext with each private key in the vector.
The new 1 step would then be
Multiple token transactions combine well with the use of one key for many tokens as discussed above, since this might reduce the number of signatures needed.
