ECE 646 Fall 2009 Final Exam December 15, Multiple-choice test

Transcription

1 ECE 646 Fall 2009 Final Exam December 15, 2009 Multiple-choice test 1. (1 pt) Parallel processing can be used to speed up the following cryptographic transformations (please note that multiple answers can be correct) A. DES decryption in the CBC mode B. DESX decryption in the OFB mode with j=32 C. Triple DES decryption in the CFB mode D. RSA decryption using CRT with the left-to-right modular exponentiation E. RSA encryption in the CBC mode with the left-to-right modular exponentiation F. MAC generation using HMAC with SHA-512 and a 128-bit key Please specify your assumptions, if you think that your answer may depend on any additional factors not addressed in the text of the problem. 2. (1 pt) Match the following transformations into four pairs of transformations, such that the transformations within each pair have approximately the same complexity of the best known attack: A. HMAC with SHA-512 and a 128-bit key B. RSA encryption with a bit key C. SHA-384 D. RSA signature with SHA-256 and the RSA key size 1024-bits E. Diffie-Hellman key agreement with a 1024-bit key F. RSA signature with SHA-384 and the RSA key size bits G. RC5 32/12/16 H. HMAC with SHA-256 and a 256-bit key 3. (1 pt) The smallest value of e, such that {e, N} is a valid RSA public key, AND {d, P=7, Q=11} is a corresponding RSA private key, AND the number of messages not concealed by RSA is equal to 9, is equal to: A. 3 B. 5 C. 7 D. 13 E. 17 F. other

2 4. (1 pt) The ciphertext character frequency shown in the attached figure most likely corresponds to the following cipher A. Vigenere cipher with the period d=2 B. shift cipher with the key k=2 C. transposition cipher D. affine cipher with the key (k 1, k 2 ), where k 1 =1 and k 2 =24 E. Playfair cipher F. Enigma Standard English character frequency Ciphertext character frequency

3 5. (1 pt) The number of bases a (1 a n) for which the Fermat's probabilistic primality test returns the result 'probably prime' for the Carmichael number n = 561 = is equal to: A. 1 B. 241 C. 320 D. 560 E. 561 F. other 6. (1 pt) The ratio of an average amount of numbers that need to be tested using the trial division followed by the Miller-Rabin test in order to find a prime of the size 4096 bits to the amount of numbers that need to be tested using the same tests to find a prime of the size 1024 bits is approximately equal to A. 2 B. 4 C. 8 D. 16 E. ln 4096 / ln (1 pt) Which of the following cryptographic algorithms are FIPS compliant, i.e., recognized by NIST as official U.S. government standards (more than one answer might be correct): A. DESX B. RC5 C. MD-5 D. HMAC E. RSA signature scheme F. Triple DES G. SHA (1 pt) 4096-bit RSA can be securely used in the following block cipher modes (choose as many as appropriate): A. CBC B. counter mode with j=64 C. CFB with j=4096 D. ECB E. OFB with j=8

4 Short problems Choose 4 out of 5 problems 1. (3 pts) Solve the following system of two equations with two unknowns, x and y, in the arithmetic modulo 39, i.e., find ALL possible values of pairs (x, y), for which both equations hold. 26 x + y 24 (mod 39) 2 x + y 9 (mod 39) 2. (3 pts) Determine the exact values of all messages not concealed by encryption with the following RSA public key: {e=3, N=85}. Verify your computations for at least TWO messages different than 1, 0, and N (3 pts) Put the following transformations in order according to their: a. execution time in software assuming that the program is performed on a microprocessor with a single core (start from the transformation that takes the smallest amount of time) b. execution time in hardware assuming that the hardware is optimized for maximum speed rather than minimum area (start from the transformation that takes the smallest amount of time) Justify your answer by deriving the necessary formulas. Formulate all necessary assumptions. Assume that standard padding techniques are used for encryption and signature generation. A. RSA signature generation using CRT with the 3072-bit d, 1024-bit p, and 2048-bit q. B. RSA signature verification with e= , and the modulus N length 8192 bits. C. RSA decryption using CRT with the 3072-bit d, 1536-bit p, and 1536-bit q. D. Single iteration of the Miller-Rabin test for a 1536-bit n. E. Single iteration of the Fermat test for a 3072-bit n. 4. Decrypt the following RSA ciphertext: C=48, using Chinese Remainder Theorem. Assume the following values of the RSA keys: public key = {e=7, N=55}, private key = {d=23, P=5, Q=11}. Show all intermediate results. 5. (3 pts) Determine an average total time necessary to find a prime number of the size of 600 bits, in software, with the probability of error smaller than 2-120, under the following assumptions: a. random search is used to screen through the candidate odd numbers n in the range < n < b. each odd number is tested first using a trial division by all primes smaller than B=5000; the time of the trial division by all small primes in this set is equal (on average) to t D = 1 µs

5 c. each number not rejected using the trial division is tested using the Miller-Rabin test. d. The time of a single modular multiplication of numbers of the size 600-bits is t MULMOD (k=600) = 3 µs.