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1. (text 3.8) Why is a DES weak key its own inverse? (Hint: DES encryption and decryption are the same once the per-round keys are generated.
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2. (text 5.1) Would it be reasonable to compute an RSA signature on a long message m by signing m mod-n (i.e., using (m mod-n)d mod-n as the signature).
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3. (text 5.6) Why do MD4, MD5, and SHA-1 require padding of messages that are already a multiple of 512-bits?
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4. (text 6.3) In RSA, is it possible for more than one d to work with a given e, p, and q?
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5. (text 6.8) Given your RSA signature on m1 and m2, how can one compute your signature on m1j×m2k for any positive integers j and k.
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6. Using the efficient algorithm, compute 13125 mod-15.
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7. (text 7.1) If m and n are two positive integers, show that m/gcd(m,n) and n/gcd(m,n) are relatively prime.
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8. (text 7.10) If n = p1a1 × p2a2 × × × pkak where pi is prime, what is f(n).
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9. Find all the square roots mod-15 of 1, i.e., every x in Z15 such that x×x mod-15 = 1.
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10. Find all the square roots mod-24 of 1.
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11. Given positive integers z1, z2, z3, x1, x2, x3, such that z1, z2, z3 are relatively prime, obtain a formula that yields a number x in Zz1×z2×z3 such that
x mod-z1 = x1
x mod-z2 = x2
x mod-z3 = x3
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