The modulus of RSA has been enlarged over the years in order to thwart improved attacks. As one would assume, public-key algorithms become slower as the modulus length increases. We study the relation between modulus length and performance in this problem. The performance of RSA, and of almost any other public-key algorithm, is dependent on how fast modulo exponentiation with large numbers can be performed.1. Assume that one modulo multiplication or squaring with k-bit numbers takes c · k2 clock cycles, where c is a constant. How much slower is RSA encryption/decryption with 1024 bits compared to RSA with 512 bits on average? Only consider the encryption/decryption itself with an exponent of full length and the square-and-multiply algorithm.

2. In practice, the Karatsuba algorithm, which has an asymptotical complexity that is proportional to klog2 3, is often used for long number multiplication in cryptography. Assume that this more advanced technique requires clock cycles for multiplication or squaring where c’ is a constant. What is the ratio between RSA encryption with 1024 bit and RSA with 512 bit if the Karatsuba algorithm is used in both cases? Again, assume that full-length exponents are being used.

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Consider a slotted aloha system, where the time slot equals the fixed duration of each packet. assume that there are 4 stations a, b,c, d sharing the medium. (a) stations a, b,c, d receive one packet each from higher layers at times 1.3, 1.5, 2.6,5.7 respectively. show which transmissions take place when, according to the slottedaloha protocol; describe all transmissions until all four packets have been successful. when needed, each station has access to the following sequence of random number, provided by a random number generator and drawn uniformly between 0 and 1: (1) station a draws numbers: 0.31, 0.27, 0.78, 0.9, 0.9, 0.11, 0. (2) station b draws numbers: 0.45, 0.28, 0.11, 0.83, 0.37, 0.22, 0. (3)station c draws numbers: 0.1, 0.2, 0.3, 0.4, 0. (4) station d draws numbers: 0.36, 0.77, 0.9, 0.1, 0.1, 0.1, 0.1, 0. (b) in slotted aloha, a station transmits in each time slot with a given probability. what probabilities would you assign to each of the four stations so as to: (i) maximize the efficiency of the protocol? (ii) maximize fairness among the four stations? (c) will the efficiency increase or decrease if we modify slotted aloha as follows: (i) get rid of slots and allow stations to transmit immediately? (ii) implement carrier sensing? (iii) implement collision detection? (iv) implement collision avoidance?

Three out of five seniors remain undecided about a college major at the end of their senior year.