 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 163802 students have viewed full stepbystep solutions from this chapter. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095.

Cross product u xv in R3:
Vector perpendicular to u and v, length Ilullllvlll sin el = area of parallelogram, u x v = "determinant" of [i j k; UI U2 U3; VI V2 V3].

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Eigenvalue A and eigenvector x.
Ax = AX with x#O so det(A  AI) = o.

Elimination matrix = Elementary matrix Eij.
The identity matrix with an extra eij in the i, j entry (i # j). Then Eij A subtracts eij times row j of A from row i.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Kirchhoff's Laws.
Current Law: net current (in minus out) is zero at each node. Voltage Law: Potential differences (voltage drops) add to zero around any closed loop.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

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

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Multiplier eij.
The pivot row j is multiplied by eij and subtracted from row i to eliminate the i, j entry: eij = (entry to eliminate) / (jth pivot).

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)·(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Pseudoinverse A+ (MoorePenrose inverse).
The n by m matrix that "inverts" A from column space back to row space, with N(A+) = N(AT). A+ A and AA+ are the projection matrices onto the row space and column space. Rank(A +) = rank(A).

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

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

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.