 4.6.1E: Encrypt the message DO NOT PASS GO by translating the letters into ...
 4.6.2E: Encrypt the message .STOP POLLUTION by translating the letters into...
 4.6.3E: Encrypt the message WATCH YOUR STEP by translating the letters into...
 4.6.4E: Decrypt these messages that were encrypted using the Caesar cipher....
 4.6.5E: Decrypt these messages encrypted using the shift cipher f(p) = (p+ ...
 4.6.6E: Suppose that when a long string of text is encrypted using a shift ...
 4.6.7E: Suppose that when a string of English text is encrypted using a shi...
 4.6.8E: Suppose that the ciphertext DVE CFMV KF NFEUVI, REU KYRK ZJ KYV JVV...
 4.6.9E: Suppose that the ciphertext ERC WYJJMGMIRXPC EHZERGIH XIGLRSPSKC MW...
 4.6.10E: Determine whether there is a key for which the enciphering function...
 4.6.11E: What is the decryption function for an affine cipher if the encrypt...
 4.6.12E: Find all pairs of integers keys (a, b) for affine ciphers for which...
 4.6.13E: Suppose that the most common letter and the second most common lett...
 4.6.14E: Encrypt the message GRIZZLY BEARS using blocks of live letters and ...
 4.6.15E: Decrypt the message EABW EFRO ATMR ASIN which is the ciphertext pro...
 4.6.16E: Suppose that you know that a ciphertext was produced by encrypting ...
 4.6.17E: Suppose you have intercepted a ciphertext message and when you dete...
 4.6.18E: Use the Vigenère cipher with key BLUE to encrypt the message SNOWFALL.
 4.6.19E: The ciphertext OIKYWVHBX was produced by encrypting a plaintext mes...
 4.6.20E: Express the Vigenère cipher as a cryptosystem.To break a Vigenère c...
 4.6.21E: Suppose that when a long string of text is encrypted using a Vigenè...
 4.6.22E: Once the length of the key string of a Vigenère cipher is known, ex...
 4.6.23E: Show that we can easily factor n when we know that n is the product...
 4.6.24E: Encrypt the message ATTACK using the RSA system with n = 43 · 59 an...
 4.6.25E: Encrypt the message UPLOAD using the RSA system with n = 53 · 61 an...
 4.6.26E: What is the original message encrypted using the RSA system with n ...
 4.6.27E: What is the original message encrypted using the RSA system with n ...
 4.6.28E: Suppose that (n,e) is an RSA encryption key, with n = pq are largin...
 4.6.29E: Describe the steps that Alice and Bob follow when they use the Diff...
 4.6.30E: Describe the steps that Alice and Bob follow when they use the Diff...
 4.6.31E: Alice wants to send to all her friends, including Bob. the message ...
 4.6.32E: Alice wants to send to Bob the message “BUY NOW” so that he knows t...
 4.6.33E: We describe a basic key exchange protocol using private key cryptog...
Solutions for Chapter 4.6: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 4.6
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7. Chapter 4.6 includes 33 full stepbystep solutions. Since 33 problems in chapter 4.6 have been answered, more than 133613 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Column space C (A) =
space of all combinations of the columns of A.

Fibonacci numbers
0,1,1,2,3,5, ... satisfy Fn = Fnl + Fn 2 = (A7 A~)I()q A2). Growth rate Al = (1 + .J5) 12 is the largest eigenvalue of the Fibonacci matrix [ } A].

GaussJordan method.
Invert A by row operations on [A I] to reach [I AI].

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Inverse matrix AI.
Square matrix with AI A = I and AAl = I. No inverse if det A = 0 and rank(A) < n and Ax = 0 for a nonzero vector x. The inverses of AB and AT are B1 AI and (AI)T. Cofactor formula (Al)ij = Cji! detA.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Left nullspace N (AT).
Nullspace of AT = "left nullspace" of A because y T A = OT.

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Outer product uv T
= column times row = rank one matrix.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Schwarz inequality
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.

Skewsymmetric matrix K.
The transpose is K, since Kij = Kji. Eigenvalues are pure imaginary, eigenvectors are orthogonal, eKt is an orthogonal matrix.

Subspace S of V.
Any vector space inside V, including V and Z = {zero vector only}.

Transpose matrix AT.
Entries AL = Ajj. AT is n by In, AT A is square, symmetric, positive semidefinite. The transposes of AB and AI are BT AT and (AT)I.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).