- communication security is extremely important in computer networksand one way many network protocols achieve security is to encrypt messages typical cryptographic schemes for the secure transmission of messages over such networks are based on the fact that no efficient algorithms are known for factoring large integers henceif we can represent secret message by large prime number pwe can transmitover the networkthe number qwhere is another large prime number that acts as the encryption key an eavesdropper who obtains the transmitted number on the network would have to factor in order to figure out the secret message using factoring to figure out message is very difficult without knowing the encryption key to understand whyconsider the following naive factoring algorithmfor in range( , )if = return the secret message is pif divides suppose that the eavesdropper uses the above algorithm and has computer that can carry out in microsecond ( millionth of seconda division between two integers of up to bits each give an estimate of the time that it will take in the worst case to decipher the secret message if the transmitted message has bits what is the worst-case time complexity of the above algorithmsince the input to the algorithm is just one large number rassume that the input size is the number of bytes needed to store rthat isn (log )/ and that each division takes time (nc- sequence contains unique integers in the range [ ]that isthere is one number from this range that is not in design an ( )time algorithm for finding that number you are only allowed to use ( additional space besides the sequence itself - al says he can prove that all sheep in flock are the same colorbase caseone sheep it is clearly the same color as itself induction stepa flock of sheep take sheepaout the remaining are all the same color by induction now put sheep back in and take out different sheepb by inductionthe sheep (now with aare all the same color thereforeall the sheep in the flock are the same color what is wrong with al' "justification" - let be set of lines in the plane such that no two are parallel and no three meet in the same point showby inductionthat the lines in determine th( intersection points