tEG$=45$

The following ten messages were signed by some of the eight participants.

$1.$No promotion $2.$Invest now $3.$Find another $4.$Buy Apple $5.$Buy HP $6.$Hire Zoran $7.$No profits $8.$Wait some $9.$Compute it $10.$Sell Cisco

Here, in no particular order, are the signatures on these messages.

y$[1]=412041295992123020240352453898046468$ s$[1]=108722729425282408579460067715408580$ y$[2]=731677238858704244758259298751682281$ s$[2]=896326910580954972205555565958385558$ y$[3]=666817389136838952618891691365765752$ s$[3]=863071972573729162183876106087410811$ y$[4]=816586613921642041366383398347476703$ s$[4]=194501397874859909804911148415489345$ y$[5]=219053362914589476320927331455302923$ s$[5]=678153544331677211475299051583835159$ y$[6]=363400839886872826522107008947036877$ s$[6]=382287219989846890283203042484007465$ y$[7]=810062756926377262816573489441310814$ s$[7]=429150489277081157437859679679303226$ y$[8]=455083728685039681079056638582289507$ s$[8]=138812955655581234882658124139814785$ y$[9]=412041295992123020240352453898046468$ s$[9]=792837565951514253278213880944803693$ y$[10]=200538392268786826989428966213397620$ s$[10]=523132019370447562981241099088063395$

A Legendre type covert channel was installed in the signature software used by some of

the users. The covert channel leaks private keys of bitlength at most $40$ bits in blocks of

five bits at a time. It came to your attention that the following eight prime numbers are used

in the same way as the $160-$bit channel featured in the course to leak the private key.

p$[1]=6752772359$ p$[2]=5821713743$ p$[3]=9811641793$ p$[4]=605040017$ p$[5]=7325468173$ p$[6]=6014097223$ p$[7]=6471525877$ p$[8]=8185301261$

PART A$.$ Find the private decryption key for each of the eight.