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Solutions of Exam I

# Solutions of Exam I - 1(4 pt Increasing the key length is...

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1. (4 pt) Increasing the key length is one way to increase security of an encryption algorithm against the brute-force type of attack. DES uses 56-bit key, which is not secure, given the modern computing power. Assume that 56-bit key was just sufficient in 1979 when DES was standardized, and assume that the hardware performance improves about 40% per year, then how many bits of a DES key would just suffice this year? Until what year would a 112-bit DES key be sufficient? Answer: 40% improvement per year doubles the performance every two years (1.4×1.4 2). So keys must grow by about 1 bit every two years. (2008-1979)/2+56=70.5, so we need 71 bit keys which would just suffice this year. (x-1979)/2+56=112, x=2091; 112-bit key would suffice till year 2091. 2. (6 pt) Suppose you encrypt message m by computing . How do you decrypt? 13 mod 77 c m Answer: (77) (7 11) (7 1)(11 1) 60 ϕ ϕ = × = = 13 is relatively prime to 60, 1 13 mod 60 d = Show steps of extended Euclidean algorithm to find d= -23 mod 60 or 37 mod 60 = 37 13 13 0 60 8 60 4 13 5 13 1 8 3 8 1 5 2 5 1 3 1 3 1 2 = × = × = − × = − × = − × = − × ՜ 1 3 1 2 3 1 (5 1 3) 2 3 1 5 2 (8 1 5) 1 5 2 8 3 5 2 8 3 (13 1 8) 5 8 3 13 5 (60 4 13) 3 13 5 60 23 13 = − × = − × − × = × − × = × − × − × = × × = × × − × = × × = × × × = ×

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