- 46.46.1: Suppose n D 589 D 19 31 and let e D 53. Bobs encryption function is...
- 46.46.2: Suppose n D 589 D 19 31 and let d D 53. Bobs decryption function is...
- 46.46.3: Suppose Bobs encryption function is E.M / D M53 .mod 589/. Alice en...
- 46.46.4: The integer n D 3312997 is the product of two distinct primes. Use ...
- 46.46.5: The first step in all public-key cryptosystems is to convert the En...
- 46.46.6: Suppose Bob creates two RSA encryption algorithms as follows: First...
- 46.46.7: Let Bobs encryption function be E.M / D Me mod n where n D pq for d...
Solutions for Chapter 46: Public Key Cryptography III: RSA
Full solutions for Mathematics: A Discrete Introduction | 3rd Edition
Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).
A = CTC = (L.J]))(L.J]))T for positive definite A.
Column space C (A) =
space of all combinations of the columns of A.
Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.
Eigenvalue A and eigenvector x.
Ax = AX with x#-O so det(A - AI) = o.
Invert A by row operations on [A I] to reach [I A-I].
Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.
Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
Linear transformation T.
Each vector V in the input space transforms to T (v) in the output space, and linearity requires T(cv + dw) = c T(v) + d T(w). Examples: Matrix multiplication A v, differentiation and integration in function space.
Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.
Ps = pascal(n) = the symmetric matrix with binomial entries (i1~;2). Ps = PL Pu all contain Pascal's triangle with det = 1 (see Pascal in the index).
The diagonal entry (first nonzero) at the time when a row is used in elimination.
Projection matrix P onto subspace S.
Projection p = P b is the closest point to b in S, error e = b - Pb is perpendicularto S. p 2 = P = pT, eigenvalues are 1 or 0, eigenvectors are in S or S...L. If columns of A = basis for S then P = A (AT A) -1 AT.
Random matrix rand(n) or randn(n).
MATLAB creates a matrix with random entries, uniformly distributed on [0 1] for rand and standard normal distribution for randn.
Rayleigh quotient q (x) = X T Ax I x T x for symmetric A: Amin < q (x) < Amax.
Those extremes are reached at the eigenvectors x for Amin(A) and Amax(A).
Right inverse A+.
If A has full row rank m, then A+ = AT(AAT)-l has AA+ = 1m.
Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
Iv·wl < IIvll IIwll.Then IvTAwl2 < (vT Av)(wT Aw) for pos def A.
v + w = (VI + WI, ... , Vn + Wn ) = diagonal of parallelogram.
Vector space V.
Set of vectors such that all combinations cv + d w remain within V. Eight required rules are given in Section 3.1 for scalars c, d and vectors v, w.