$(1)(4$ pts$)$ Money is exchanged in Bitcoin $($a digital currency system$)$ through transactions. A simplified description of how it works is given in the figure below. $($i$)$ Each transaction has a certain number of $$inputs$$ and $$outputs$($see the left figure below$).($ii$)$ The name in an output represents the recipient of the money in the transaction, and the amount represents the amount given to that recipient. To simplify the discussion, we assume there is only one recipient $($output$)$ in each transaction. $($iii$)$ The inputs are just pointers pointing to $$outputs$$ of some previous transactions which contain the money owned by the person who generates the current transaction $($e$.$g$.$ transaction W in the figure below$).$ In the example shown in the left figure, Eric sends $$50$ to Alex. So he generates transaction W and the two inputs of W point to the outputs of two previous transactions X and Y in which Eric is the recipient. In Bitcoin, however, all transactions are in the public domain and visible to everyone. In order to keep users anonymous, each user creates a $($public$,$ private$)$ key pair based on a given public$-$key cipher and uses the public key as his$/$her identity. The private key, of course, should be kept as a secret. No names are used in the system. So the concept of certificate for binding a public key and its owner$$s name does NOT apply here. Let Eric$$s $($public$,$ private$)$ key be denoted by $(+,),$ and Alex$$s by $(+,).$ So $$Eric$$ and $$Alex$$ in the left figure below will be replaced by $1+$ and $2+$ in a real transaction $($see the right figure below$).$ But anonymity causes a problem. When Alex receives transaction W$,$ how can Alex be sure the sender indeed is the owner of $1+($note: anybody can claim to be the owner of $1+)($a$)$ What Eric needs to attach to Transaction W to show he is indeed the owner of $1+?($b$)$ How can Alex verify that the sender indeed is the owner of $1+?$