# Massachusetts Institute of Technology Handout : Network and Computer Security October 9, 2003 Professor Ronald L. Rivest.

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3 Handout 13: Quiz 1 Name: 3 (e) Describe how to efficiently test if g is a generator modulo a prime p, where p 1 = 4q 2 for some prime q > 2. Solution: Check that g x 1 (mod p), where x is every divisor of 4q 2 (other than 4q 2 itself): x = 1, 2, 4, q, q 2, 2q, 4q, 2q 2. If all tests pass, g is a generator. In fact, it is sufficient to do the test only for x = 4q, 2q 2, because if any of the above tests fail, they will also fail for one of those two values of x. (f) The year is 2050 and Moore s law has continued to hold, doubling computer performance every 18 months. Is it time to retire AES-128? How about AES-192 or AES-256? Explain your reasoning. Solution: Today, 80 bit symmetric keys offer only decent security. In our scenario of the future, computers will be about 2 31 times faster, implying that 111 bit keys are questionable. At this point, the security of AES-128 is in question (only one student thought about the lifetime of the data being protected; it may be time to retire the cipher if the data needs to stay secure long after 2050, which is often the case). AES-192 and AES-256 are still fine. Of course, not many people believe that Moore s Law will hold for so long...

4 Handout 13: Quiz 1 Name: 4 Problem Q1-2. Security Policy [9 points] For each of the 9 items listed on the front page of this quiz, choose whether the item contains elements of a security policy, security mechanism, both, or neither. Check one box per row. Item Policy Mechanism Both Neither 1 x 2 x 3 x 4 x x 5 x 6 x 7 x 8 x 9 x Solution Note: We accepted either mechanism or neither for item 4.

5 Handout 13: Quiz 1 Name: 5 Problem Q1-3. True or False [36 points] Circle T or F for each of the following statements. No justification is required, but if you think the question is ambiguous, state your clarifying assumptions. True False In the Fermat primality test, there is a small (but non-zero) probability that the candidate number will be declared composite when it is actually prime. Solution: False. The one-sidedness of the error goes the other way. True False A deterministic public-key encryption algorithm can never be secure against an adaptive chosen-ciphertext attack. Solution: True. See the lecture defining the attack. True False A PGP public key is more trustworthy if it has been signed by a lot of other public keys. Solution: False. First, you shouldn t trust signatures from keys you haven t authenticated. Second, even if you believe the key is valid (authenticated), you shouldn t necessary trust it to properly sign other keys. True False A good way to generate an AES encryption key is to apply SHA-1 to the number of seconds that have elapsed since January 1st, Solution: False. The domain is too small (only about 2 30 seconds have elapsed since the epoch). Additionally, if the adversary knows the approximate time at which the key was created (which is almost certainly true), the search space is reduced significantly.

6 Handout 13: Quiz 1 Name: 6 True False It is possible to compute the SHA-1 value of every number in the range 1,..., 10 10, on today s standard PC, in a single day. Solution: True. One can invoke SHA-1 over 200, 000 = times per second, so it would take 50, 000 seconds, or about 14 hours, to complete the given task. True False It is not known if computing φ(n) is harder than computing the prime factors of N, where N is an RSA modulus. Solution: False. The problems are known to be equivalent: if you can find primes p, q such that N = pq, then φ(n) = (p 1)(q 1). Conversely, if φ(n) is known, then p + q = N + 1 φ(n), and pq = N. Finding p and q is now just a matter of solving this quadratic equation. True False In RSA, for a given modulus N, any prime larger than 2 may be used as the public exponent e. Solution: False: e must be relatively prime to φ(n), so any e that divides φ(n) cannot be used. True False It is possible to have a digital signature scheme where forgery is impossible for a computationally unbounded adversary. Solution: False. Signatures are publicly verifiable, therefore an unbounded adversary can simply try all signatures until she finds one that verifies correctly. True False If p = 2q + 1 where p and q are primes, then 1 is not a square mod p. Solution: True (sort of). Whenever safe primes are used, q is meant to be an odd prime, but we did not make this explicit. A number a is a square mod p if and only if a q = 1 (mod p). Because q is odd, ( 1) q = 1 (mod p), so 1 is not a square. We gave full credit to anyone who pointed out the (only!) counterexample of q = 2.

7 Handout 13: Quiz 1 Name: 7 True False Given an AES key K, it is possible to efficiently construct a one-kilobyte plaintext message M such that an encryption of M under AES (using key K) in CBC mode consists of all 0s. Solution: True. Simply build the ciphertext of 1024 bytes (plus 128 bits for the IV), and decrypt it under CBC mode with key K. The result is a message which can encrypt to all 0s. True False Using a one-time pad for encryption, followed by a CBC-MAC on the resulting ciphertext, provides unconditional secrecy and computationally secure authentication. Solution: True. No secrecy is compromised, because the message is encrypted before the MAC is performed. True False When k is large, testing a randomly-generated k-bit number t for primality can be done for use in a cryptosystem by testing whether 2 t 2 (mod t). Solution: True. The chance that a randomly-generated number (of the size used in cryptosystems) is composite, but also a base-2 pseudoprime, is at most The risk of this event is negligible, and in any case, the first ciphertext probably won t decrypt properly, alerting the key owner to the problem. True False If h( ) is collision-resistant, then h(h( )) is collision-resistant. Solution: True. If it is feasible to find a collision x, x under h(h( )), then either x, x is also a collision under h, of h(x) h(x ) is a collision under h. In any case, h is not collision-resistant, which is a contradiction.

8 Handout 13: Quiz 1 Name: 8 True False SHA-1 is more resistant to attacks based on the birthday paradox than MD5. Solution: True. The output of SHA-1 is 160 bits, whereas the output of MD5 is only 128 bits. True False Moore s law implies that you should change your password once every 18 months. Solution: False. If your password is very long to begin with, then there may be no reason to change it for a long time. True False The perfect security of a one-time MAC means that an attacker can never guess the MAC for any given message. Solution: False. It simply means that the adversary chance of guessing the MAC is 1/p, where p is the range of the MAC. There is still a non-zero chance of a correct guess. True False The design of the Feistel network requires that the round function F be invertible. (F is the function such that R = L F (R).) Solution: False. The design of the Feistel network allows one to decrypt, even when F is not invertible. True False Define h(x) = g (x2 ) (mod p), where p is a large prime and g is a generator mod p. The function h is collision-resistant. Solution: False. Both 1 and 1 hash to the same value.

9 Handout 13: Quiz 1 Name: 9 Problem Q1-4. Multiple Choice [12 points] (a) If an efficient algorithm for calculating discrete logs over an integer modulus is discovered, which of the following schemes will be known to be not secure (circle all that apply): I AES II Diffie-Hellman III RSA IV El Gamal Solution Note: RSA will be insecure, because given a plaintext/ciphertext pair (m, c = m e (mod N)) = (c d (mod N), c) (which the adversary can compute himself), one can discover the secret decryption key d by computing the mod N discrete log of m, base c. (b) If an efficient algorithm for factoring large numbers is discovered, which of the following schemes will be known to be not secure (circle all that apply): I AES II Diffie-Hellman III RSA IV El Gamal (c) A hash ring for a hash function h is a set of values v 1,..., v t such that v i+1 = h(v i ) for 1 i t 1, and v 1 = h(v t ). (Two hash rings are the same if they have the same values listed in a different order.) How many hash rings does SHA-1 have? Pick the most specific, correct answer. I II 0 III 1 Unknown IV 2 V 1 or more (nobody knows which) VI 2 or more (nobody knows which) Solution Note: There must be at least 1 hash ring, because if we start with any 160-bit input and iteratively hash, we must get a repeated value at some point. However, there may only 1 hash ring (e.g., if the ring contains all hash values), or there may be more.

10 Handout 13: Quiz 1 Name: 10 Problem Q1-5. Hot Poker [10 points] For his final project, Ben Bitdiddle decides to implement a poker server, so that he can play friendly games with his chum(p)s across campus. The process works as follows: the server deals five cards (selected at random from a deck of 52 cards) to each player as their hand, removing each card from the deck as it is dealt. The cards are known to their owner but must remain secret to the other players. Each client then sends the values of their bet to the server. Finally, the server broadcasts each client s cards, and the best hand wins (for this problem, it is not necessary to know anything about the rankings of poker hands). This is the conclusion of the hand. The process is then repeated with a new, complete deck until Ben and his friends get tired of playing. For security, Ben decides to use some encryption. Client i keeps a long-lived secret AES key K i, which the server also knows. During the dealing phase of the game, the server sends AES Ki (C) (using ECB mode) to client i for each card C in that client s hand (a card is represented in some canonical way by an integer in the range 0,..., 51). When the cards are broadcast at the end of the hand, the server sends them in the clear to all clients. (a) Explain one attack that can be executed by a passive malicious player who is able to eavesdrop on the network. Solution: Since each card is sent as a single block with ECB encryption, there is a one-to-one mapping between the ciphertexts and the plaintexts. The passive malicious player can watch the encrypted cards go by and build up a codebook mapping these encrypted blocks to unencrypted cards. (b) Suggest a fix to this problem. You get an extra point if your fix does not increase the length of the messages from the server to the clients. Solution: Since each card can be encoded as a 6-bit value, and since the AES block size is 128 bits, the simplest solution is for the dealer to encrypt C R where R is a random 122-bit block. The player software would automatically remove the nonce. Other acceptable answers were to use CBC mode. We liked CBC answers that specified how the IV was set for example, by using the hand number as the IV. Yet another acceptable answer was to specify a key schedule for the encryption key. For example, specifying that each key K i = SHA-1(K i 1 ). We took off a point for students who suggested using Diffie-Helman for exchanging new AES keys each hand. This works, but it is extremely inefficient relative to the original scheme.

11 Handout 13: Quiz 1 Name: 11 Problem Q1-6. Tripwire [10 points] Tripwire is a program that is used to assure the integrity of the files on a Windows or Unix workstation. In one incarnation, the Tripwire executable uses the computer s cryptographic library to compute the MD5 hash code for every file in directories that are specified by the Tripwire configuration file. Tripwire then creates a database that consists of each file s name and MD5 code. This database is stored on the computer s hard disk. Ben configures Tripwire to compute the MD5 values of all the programs in his /bin and /usr/bin directories. A month later, he runs the Tripwire program again to see if any of the MD5 codes have changed. The database that is generated perfectly matches the database of MD5 values generated when Ben first ran Tripwire. But unbeknownst to Ben, his files have been compromised by an attacker. Describe three attacks that are consistent with this scenario, and how you would defend against them. Solution: There were many correct answers, including: Change the database, so that it contains the MD5 values of the modified files, rather than the original files. Change the function that calculates MD5, so that it returns the old values, rather than the new ones. Hack the kernel so that the Tripwire program thinks it is reading the files in the /bin and /usr/bin directories, when it is in fact reading other files (presumably the original copies). Hack the Tripwire program, so that it says that nothing is wrong, when in fact something is. Hack Ben s operating system, so that Ben sees the Tripwire program report that everything is okay, when in fact it is reporting something else. One answer was not accepted. Students did not receive credit for arguing that MD5 is weak and that the attacker could simply modify the new files so that they have the same MD5 as the original files. Any student who lost credit for this answer is invited to produce two different files that have the same MD5 hash code.

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