CSE5473 Homework 1 (due Wednesday, February 11) 0. A disadvantage of the general monoalphabetic cipher is that both sender and receiver must commit the permuted cipher sequence to memory.A common technique for avoiding this is to use a keyword from which the cipher sequence can be generated. For example, using the keyword CIPHER, write out the keyword followed by unused letters in normal order and match this against the plaintext letters: plain: a b c d e f g h i j k l m n o p q r s t u v w x y z cipher: C I P H E R A B D F G J K L M N O Q S T U V W X Y Z If it is felt that this process does not produce sufficient mixing, write the remaining letters on successive lines and then generate the sequence by reading down the columns: C A K S Y I B L T Z P D M U H F N V E G O W R J Q X This yields the sequence: C A K S Y I B L T Z P D M U H F N V E G O W R J Q X Such a system is used in the following example. The ciphertext is: UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ The plaintext is: it was disclosed yesterday that several informal but direct contacts have been made with political representatives of the viet cong in moscow Determine the keyword. 1 1. Alternating ones + zeros (0101010101010101...01) is one of DES weak keys. Give a detailed proof that this DES weak key is its own inverse. (Hint: DES encryption and decryption are the same once the per-round keys are generated.) 2. To speed up RSA, we can require that e=3 for all users. This creates the possibility of a cube root attack. Explain this attack and how to prevent it. For (N,e)=(33,3) and d=7, verify that the cube root attack works when M=3 but not when M=4. 3. Consider the cryptographically secure pseudo random generator with a pseudo random function f in figure 1 and then answer the following questions. Figure 1: Cryptographiically secure PRGs (1) Is the sequence of keys K0, K1, K2... random? (2) Can this generator be used for providing key sequence to a stream cipher? If so, what property of it implys that it can be used? 4. Answer the following questions about a family of 6 people, assuming that birthdays are uniformly distributed through the days of a week, month and a year. Also assume that the year is exactly 365 days and each month is exactly 30 days. a. What is the probability that at least two of the family members have the same birthday? and what is the probability that none of them have the same birthday? b. What is the probability that at least two of the family members are born 2 in the same month? and that none of them were born in the same month? c. What is the probability that at least one of them is born on the first day of a month? 5. Let π be a permutation of the integers 0, 1, 2, ... (2n - 1) such that π(m) gives the permuted value of m, 0≤m<2n . Put another way, π maps the set of n-bit integers into itself and no two integers map into the same integer. DES is such a permutation for 64-bit integers. We say that π has a fixed point at m if π(m) = m. That is, if π is an encryption mapping, then a fixed point corresponds to a message that encrypts to itself. We are interested in the probability that π has no fixed points. Show the somewhat unexpected result that over 60% of mappings will have at least one fixed point. General Notes. In addition to the readings of Lecture 1, read also: 11.111.3 3