Secure Communicating Systems Design Analysis And Implementation 1st Edition Michael R. A. Huth - Solutions

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=+8-12.For the wall described in Problem 8-4, with an outside surface temperature profile given by Table 8-5 and a constant inside surface temperature of 70 F. determine the inside conduction heat flux for each hour.
=+8-11. Compute the solar irradiation for a south-facing wall in Boise, ID, for each hour of the day on July 21. Assume 0.4 percent outdoor design conditions. The wall has a solar absorptivity of 0.9.a thermal emissivity of 0.9. negligible thermal mass, an outside-surface-to-inside-surface U-factor
=+8-10.Compute the solar irradiation for a west-facing wall in Albuquerque, NM, for each hour of the day on July 21. Assume 0.4 percent outdoor design conditions. The wall has a solar absorptiv-ity of 0.8. a thermal emissivity of 0.9, negligible thermal mass, an outside-surface-to-inside-surface
=+8-9.A wall has an incident solar radiation of 300 Bu/(hr-ft-), an outside air temperature of 98 F, and an outside wind speed of 15 mph. The wall has a solar absorptivity of 0.6, a thermal emissivity of 0.9. negligible thermal mass, an outside-surface-to-inside-surface U-factor of 0.1
=+8-8.A roof is composed of asphalt roll roofing, 4 in. of 120 lb/ft limestone concrete, 2 in. of expanded polystyrene, a 3 - in. airspace, and 0.5 in. of acoustical tile. Determine the conduc-tion transfer function coefficients.
=+8-7.The roof of Problem 8-6 is changed to have a suspended ceiling with a 12 in, air space above it. Determine the conduction transfer function coefficients
=+8-6.A roof is composed of asphalt roll roofing, 1in. plywood, 5 4 in. mineral fiber insulation (R-19).and in. gypsum board. Determine the conduction transfer function coefficients.
=+8-5.Change the insulation in Problem 8-4 to R-19, and determine the conduction transfer function coefficients,
=+8-4.Determine the wall conduction transfer function coefficients for a wall composed of 4 in, brick[k = 7 (Btu-in.)/(hr-ft2-F)], in. regular density sheathing (vegetable fiber board). 34 in. min-eral fiber insulation (R-13), and ] in, gypsum board.
=+8-3.Determine the ASHRAE Standard 90.1 design conditions for the following locations, Include the maximum outdoor temperature, the outdoor mean coincident wet bulb temperature, the indoor dry bulb temperature, the relative humidity, the elevation, and the latitude. (a) Norfolk.VA. (b) Pendleton.
=+8-2.Southern coastal regions of the United States experience periods of very high humidity. Explain how this might influence selection of design conditions.
=+8-1. Describe a situation where the heat gain to the space is (a) greater than the cooling load at a given time. (b) less than the cooling load at a given time, and (e) equal to the cooling load at a given time.
1. Let Ex(): P → C and Dk(): C → P satisfy (2.1).(a) Show that the function Ex(): P → C is injective; that is, show for all M, M' εP that the equation Ex (M) = EK (M') implies M = M'.(b) Explain why and under which circumstances an encryption function should be injective.(c) Assume that P
2. Types of attacks Consider the following types of attacks on a key-dependent crypto-system. The attacker attempts to recover the key. She• possesses a sample of cipher-text (cipher-text-only attack);• has (temporary) access to the decryption function and so can choose cipher-texts and compute
1. Describe how Bob can send a signed and secure message N to Alice.
2. Think of some possible scenarios in which Alice may successfully dispute that she actually signed a message according to the protocol just discussed. What changes to this protocol can you suggest that will make it more difficult for Alice to deny the authenticity of her digital signature?
3. The boolean function $\oplus$ computes the exclusive-or of two boolean values: for v, w $\in${0, 1}, we have v $\oplus$ w = 1 if and only if v $eq$ w; otherwise, v $\oplus$ w = 0.(a) Show that (x $\oplus$ v) $\oplus$ v = x for all x, v $\in$ {0, 1}.(b) Let M be a string of length n over the
1. (a) Discuss how realistic or unrealistic the assumptions are concerning the attacker's capabilities and knowledge in Attack 2.6.(b) (i) Modify Attack 2.6 so that Mallory replaces Alice's message with one of her own choosing.(ii) Explain why Bob will be unaware of this replacement.(iii) Is
1. Explain: A number n ∈ N is not prime if and only if it has a factor a ∈ N such that 2≤ a ≤ √n + 1.
2. Show that a mod b and a divb are well-defined. That is, show that if r, r' are two numbers in {0, 1, 2, ..., b-1} and k, k' ∈ Z such that a = r+k·band a = r'+k'·b, then r = r' and k = k' follow. (Hint: Use 0 = (r-r') + (k-k')·b.)
3. Describe and implement an algorithm that uses only addition and subtraction and takes integersa, b in decimal representation as input and computes a modb. You may first address this fora, b ≥ 0.
4. Repeat the previous exercise for computing a div b.
5. Prove: If m|x and m|y, then m|(rx+sy) for all r, s ∈ Z.
6. For integers k and l such that k|l and l|k, what can you say about the relationship between k and l?
1. RSA example Let p = 7 and q = 17.(a) Compute n.(b) Lete = 5. Compute d.(c) Encrypt 49 and 12. Verify that the decryption of those resulting numbers recov-ers the original ones.(d) Decrypt 49 and 12. Verify that the encryption of those resulting numbers recov-ers the original ones.
2. We said that the set of plain-texts equals the set of cipher-texts for the RSA cryp-tosystem. However, can you think of some values in the message space that are undesirable?
3. Consider the RSA cryptosystem, where$p \stackrel{\text{def}}{=} 25525635435900842730349748303929424117$,$q \stackrel{\text{def}}{=} 259965242284515788826732110240207250949$.(a) Compute the modulus *n*.(b) Compute (*p* - 1) * (*q* - 1).(c) Which of the two possible public-key exponents *e* is
4. Discuss the difference between attacks that are dependent on a specific implementation of a cryptosystem and attacks that would work for all implementations.
1. (a) Show: n∈ N is prime if and only if all numbers n mod 2, n mod 3, ..., n mod $$\sqrt{n}$$are different from 0. Show that if the computation of *n mod k* takes constant time, then this primality test is in Θ(√*n*).(b) Show: If *n* has a binary representation with β bits, then β equals
1. Prove formally that our candidate invariant is an actual invariant for the for-statement in the sliced program Witness (a, n) in Figure 2.5. If you know about program logics, you may use one; otherwise, use a proof by mathematical induction on the value of i.
2. Let n ∈ N and x, y ∈ Z. Prove:(a) (x + y) mod n = (x mod n) + (y mod n) mod n;(b) (xy) mod n = (x mod n) (y mod n) mod n; and(c) xt mod n = (x mod n)t mod n for all t∈ N.
3. Prove the two equations in (2.11), using the facts of the previous exercise.
4. Implement the algorithm Miller-Rabin (n,s) in a programming language of your choice. Why did we not write a = Random (1,n-1) in Figure 2.2?
5. Generating primes with specific properties Often, one wants to generate primes with some specific properties. For example, you are asked to generate two large primes p and q such that:(a) the primes p and q differ in length by only a few binary digits;(b) the numbers p - 1 and q - 1 contain
6. **Pseudo-random numbers** Use an implementation of the Miller-Rabin algorithm and the technique of iterative squaring to implement the pseudo-random number gen-erator due to M. Blum and S. Micali. (Note that the Miller-Rabin algorithm already relies on some pseudo-random number generator.) The
7. Primality testing Consider the infinite list of natural numbers, where the first ele-ment of that list is 12 and the next list element, as a string, is the concatenation of the previous number-string with a digit *d*, where *d* is *e* + 1 mod 10 and *e* is the rightmost digit of the previous
8. Pseudo-random numbers L. Blum, M. Blum, and M. Shub designed a more effi-cient pseudo-random number generator with the additional property that the *i*th bit*bi* can be computed "directly" from *i* and the seed *x0*. Use Miller-Rabin(*n*,*s*) to implement it as follows.(a) Generate two large
9. Blum-Goldwasser PKC We refer to the notation and concepts of the previous ex-ercise to implement a public-key cryptosystem (due to M. Blum and S. Goldwasser)which requires computational resources comparable to those of RSA but which, un-like RSA, has the security feature that the encryption
1. Show that =n is an equivalence relation for each n ∈ N. That is, show that =n is(a) reflexive: x =n x for all x ∈Z;(b) symmetric: x =n y implies y =n x for all x, y ∈ Z; and(c) transitive: x =n y and y =n z imply x =n z for all x, y, z ∈Z.
2. (a) Show that Z is the disjoint union of sets [x]n, where x ranges over Z. Thus you need to show that Z is the union of such sets and that [x]n and [x']n are either the same or have empty intersection for any x, x' ∈Ζ.(b) Show that there are exactly n different such classes [x]n.
1. Show that the operation + : Zn x Zn → Zn is well-defined. That is, show that if x' =n x and y' =n y then the classes [x + y]n and [x' + y']n are the same.
2. Show that the operation * : Zn x Zn → Zn is well-defined. That is, if x' =n x and y' =n y, then the classes [x . y]n and [x' . y']n are the same.
3. Is 3 a member of [12]13 +13 [17]13?
4. Is 5 a member of [41]7 *7 [6]7?
1. Explain in detail why [2]6 is not a unit.
2. Recall the definition of x⊕y, the exclusive-or of x and y, from Exercise 2.2-3 (p. 20).Show that x ⊕ y = x + y mod 2 for all x, y ∈ {0, 1}.
3. Show that [160]841 is a unit in Z841 by computing an x ∈ Z with 160 ⋅ x = 1 mod 841.(Hint: Use the algorithm Extended_Euclid from Exercise 2.19-1, p. 60.)
4. Show that the following are groups.(a) (Sn, ⊕, Ô), where Sn is the set of binary strings of length n, ⊕ is the exclusive-or operation applied bitwise on such words, and Ô is the string of 0s of length n.Can you use the fact of Exercise 2 for a quick argument that the operation ⊕ must be
5. (a) Show that inverses in groups are unique. That is, if xoy = yox = e and xoy' =y'ox =e, then y = y'. Make sure that you indicate clearly which properties of groups your argument uses and at which points.(b) Use the previous item to show (2.18).
6. Show that e is the only two-sided identity in a group (G, o, e).
7. Explain why the following data are not sufficient for obtaining a group and explain in which cases you may complete the given data to obtain a group; otherwise, explain why no group could result from the given data.(a) Consider the set of finite binary strings with the operation $x \circ y =
1. Write a program that uses Proposition 2.26 to compute φ(n) for "small" values of n.
2. Prove:(a) If A is a subset of S, then |A| ≤ |S|.(b) For any subset A of S, we have |S| = |A|+|S\A|, where S\A is the set of all elements of S that are not in A.(c) For any subsets A and B of S, let A ∩ B be their intersection and A U B their union; then|A| + |B| = |A ∩ B| + |A U B|.(2.20)
3. Let (G, o,e) be a group and let H be a nonempty subset of G. Show that H is a sub-group of G if and only if a o b-1 ∈ H for alla, b ∈ H.
4. Lagrange's theorem Prove Lagrange's theorem. For each a ∈ G, define the set$$aH := {a o h | h ∈ H}.$$Show the following.(a) The set G is the union of all sets aH, where a ranges over G.(b) Fora, b ∈ G, if the sets aH and bH have a nonempty intersection then they are equal.(c) Fora, b ∈
5. Prime factorization Prove that every n ∈ N has a unique factorization$$n = p_1^{a_1} \cdot p_2^{a_2} \cdots p_k^{a_k}$$,(2.21)where the pi are prime numbers in increasing order and ai ∈ N. (Hint: Use mathe-matical induction on n. In the inductive step, argue by cases: what if n + 1 is
6. Infinitely many prime numbers Prove: There are infinitely many prime numbers.(Hint: Show that, for any n ∈ N, there is a prime number p such that n < p ≤ 1.2.3 · (n - 1) · n + 1.)
7. Let Hi be a subgroup of G for all i ∈ I. Show that the intersection of all these sub-groups, ⋂i ∈ I Hi, is also a subgroup of G.
8. Show: For g ∈ G, the subgroup (g) of a group (G, o,e) is commutative. (Is it always finite?)
9. Let T be a subset of a group (G, o, e). Show that (T) equals the intersection of all subgroups of G that contain T. Thus, (T) in Definition 2.31 is indeed well-defined.Prove Fermat's theorem (Theorem 2.15). Consider the group
10. Fermat's theorem(Z*n, *n, [1]n).(a) Show that([a]n) = {[1]n} ∪ {[a]m | m ∈ N}for any [a]n ∈ Z*n.(b) For a ∈ Z with gcd(a, n) = 1, argue that there must exist a smallest number / in[0] ∪ N such that [a]'n = [1]n.(c) Use Lagrange's theorem to conclude that there exists some k ∈ N such
11. Euler's theorem Generalize the proof of Fermat's theorem appropriately to prove Euler's theorem: For any n > 1, the number a^(φ(n)) mod n equals 1 for all a such that[a]n ∈ Z*n.
12. Let p and q be prime numbers.(a) Prove that φ(p ⋅ q) = (p - 1) ⋅ (q - 1).(b) Let n = p ⋅ q and x ∈ Z. Define x0 i ≥ 0.def= x² mod n and xi+1 = x² mod n for all(i) Use mathematical induction on *i* to show that x*i* = x 02i mod *n*.(ii) Use the facts from Exercises 11 and 12(a) to
1. Verify that 2 is a primitive root for $Z_{11}^*$.
2. Compute all primitive roots g for $Z_{11}^*$ with 0 < g < 11.
3. Determine the smallest positive primitive root g for $Z_{13}^*$ and compute log(13,g)(x) for all x ∈ {1, 2, 3, ..., 12}.
4. Find the smallest value of n > 2 such that Zp² has no primitive root.
5. Computing primitive roots In this problem you may use two mathematical facts that are demonstrated in the Appendix of this text. First, the proof of Theorem A.20(p. 268) reveals that if g is a primitive root for Zp, then it is also a primitive root for Zpk, where k is any natural number greater
6. (a) Give an upper bound on how many operations i → gi mod n one must perform in order to verify (or refute) that an invertible element g is a primitive root for Zn*(b) Assume that you have a complete prime factorization of φ(n). Using Lagrange's theorem, how many operations gi mod n do you
7. Nontrivial square roots of 1(a) Count the number of n between 1 and 1000 such that there are exactly 14 non-trivial square roots of 1 modulo n (the trivial ones are 1 and n − 1).(b) Repeat part (a) with 4 in place of 14. How many n do you count now?
1. Show that (Zo, to. [0]o) is isomorphic to (Z, +, 0).
2. Find a simply described group that is isomorphic to (Z1, +1, [0]1>.
3. Show that, if n: G → H and : H → G are isomorphisms between two groups, then they preserve inverses: n(g-1) = n(g)-1 and (h-1) = (h)-1 for all g∈ G and h∈ H.
4. Prove Theorem 2.35 as follows.(a) Show that η and ψ are well-defined.(b) Show that η preserves two-sided identities, that is, η[1] = [0]¢(n).(c) Show that ψ preserves two-sided identities, that is, ψ[0]φ(n) = [1]η.(d) Show that η preserves group multiplication, that is,η([x]η *η
1. Determine the order of the element [2]7 in the group (Z7*, *7, [1]7).
2. Prove Lemma 2.38.(a) Let * be ordG(g). Show that (g) equals {g1, g2, ..., g*}.(b) Conclude that the size of (g) is less than or equal to *.(c) Show that the set {g1, g2, ..., g*} has exactly * many elements.
3. Prove Lemma 2.39.
1. gcd recursion theorem Prove the god recursion theorem: For any a ∈ NU {0} and be N, we have gcd(a,b) = gcd(b, a mod b).(2.25)(Hint: Show that both expressions are divisors of each other. Why does this guaran-tee their equality?)
2. Euclid's algorithm Use the god recursion theorem to prove that Euclid's algo-rithm, written as a recursive program in pseudo-code below, computes gcd(a, b):Euclid(BigIntegera, BigIntegerb) {if (b== 0) return a;else return Euclid (b,a mod b);}
3. The Fibonacci numbers Fn (n∈N) are defined by F1 def 1.F2 def 1.Fn+2 def Fn+1 + Fnfor all n ∈ N.(a) Compute F7.(b) Prove: If a > b ≥ 0 are integers such that the program Euclid(a,b) makes k ∈ N recursive calls, then a ≥ Fk+2 and b ≥ Fk+1 hold. (Hint: Use mathematical induction on
4. Specify a computation trace for execution of the function call Euclid(653, 134).
5. Compute gcd(2354, 456).
6. Implement Euclid's algorithm in a programming language of your choice.
1. Use the Chinese remainder theorem to compute the unique solution of x = 2 mod 3, x = 3 mod 5, x = 2 mod 7, where 0 < x < 105.
2. Consider the Carmichael number 561.(a) Compute the set W(561). Is |W(561)| < (561 - 1)/2?(b) How many nontrivial square roots of 1 modulo 561 are there?(2.33)
3. Carmichael numbers *n* are defined via conditions on units modulo *n*. This makes sense. Show: If *a* is not a unit modulo *n*, then *a**n* - 1 mod *n* ≠ 1.
4. Compute all the Carmichael numbers that are less than one million. The three small-est Carmichael numbers are listed immediately after Definition 2.43 (p. 49); there are few such numbers below one million. The crucial part of this exercise is the program you write.
5. Let *n* = *p*1*k*1 *p*2*k*2 ... *p**m**k**m* be the unique prime factorization of *n*. Define λ(*n*) as the least common multiple (see p. 259) of all φ(*p**i**k**i*), where *i* = 1, 2, ..., *m*. Prove:(a) λ(*n*) | φ(*n*);(b) *a*λ(*n*) = 1 mod *n* for all [*a*]*n* ∈ Z*n*;(c) if *n* is a
6. (a) Write a program that takes a "small" integer *n* as input and returns the smallest Carmichael number that is greater or equal to *n*.12(b) Test your program with the inputs 323, 645, 1521, and 1999; compare your first three outputs to the three Carmichael numbers listed after Definition 2.43.
7. Prove Lemma 2.45.
8. Prove Theorem 2.46 as follows.(a) Show that the map η: Ζn → Πki=1 Ζni, withdefη[a]n = ([a]n1, ..., [a]nk)is well-defined.(2.34)(b) For mi def = n/ni, show that there exists some mi-1 ∈ Z such that [mi-1]ni is an inverse of [mi]ni in Zni*.(c) For ci def = mi ⋅ (mi-1 mod ni) ∈ Z,
9. Prove Corollary 2.47.
10. Prove Corollary 2.48.(2.35)
11. Can you apply the Chinese remainder theorem to obtain a more efficient algorithm for the decryption in (2.8) of the RSA PKC system? Could this be done for the en-cryption operation in (2.7)?
1. Let p > 1 be prime. Prove: $$p^{tp^k} - 1$$ for all k ∈ N.
2. Prove Corollary 2.51.
3. Compute an n ∈ N such that |W(n)|= (n-1)/4.
4. Factoring with square roots Let x ∈ (2, 3, ..., n-2] be a nontrivial square root of 1 modulo n. Show that gcd(x-1,n) is a factor of n. Can you even show n = gcd(x-1,n) gcd(x+1, n)?Explain how knowledge of such an x allows one to factor n efficiently.
5. Design a randomized algorithm that takes as input a Carmichael number or a prime, n, and whose output is either a factorization of n or a message saying that n is a prob-able prime. What obstacles are there if you want to extend this algorithm to apply to all natural numbers?
6. Prove or disprove: If W(n) is empty for an odd number n ∈ N, then n is prime.

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Secure Communicating Systems Design Analysis And Implementation 1st Edition Michael R. A. Huth - Solutions

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