Let 
                              "a"

 and 
                              "b"

 be real numbers and 

       
"A = Matrix(3, 3, [[a, a - 1, -b], [a - 1, a, -b], [b, b, 2*a - 

   1]])"

,  
                              "B="

 "Matrix(5, 5, [[0, a, 3, 0, a], [3, 0, 0, b, 0], [0, 1, b, 0, 

    1], [b, 0, 0, 1, 0], [0, a, 1, 0, b]])"

. 

(a) Show that if 
                            "0 <= a"

, 
                            "a <= 1"

 and 
                      "b^2 = 2*a*(1 - a)"

, then A is an orthogonal matrix with determinant equal to one. 
(b) For what values of a and b is the matrix B singular? Determine the inverse of B (for those values of a and b for which B is invertible).
 

Find all rational function solutions to the Kadomtsev-Petviashvili equation 
                         
(&PartialD;)/(&PartialD;x);

diff(u, t) + 6*u*diff(u, x) + diff(u, x, x, x) - diff(u, y, y) = 0;

by u = 2 
diff(ln, x, x)*f;

=(2 (((&PartialD;)^2)/(&PartialD;x^2) f) f-2 ((&PartialD;)/(&PartialD;x) f)^2)/(f^2);
 with 
f;
  =
(a[1 ]x+a[2] y+a[3] t+a[4])^2+(a[5] x+ a[6] y+a[7] t+a[8])^2+a[9], ;
where 
a[i], i=1..9, ;
are real constants.
 

Suppose you know that a person's public key modulus m is the following 200 digit number :
 
"166571820085905498772938593207359906487315774156514145089564134\

  67009247675308229223573508773729914408724747749233779284765414\

  62530835296763930087228227919982503096221081674037688617693027\

  1167988018683"

Factor m and check your factorization after the factors are found.  Use the following Fermat factorization method:  Note that if  
                        "m = x^2 - y^2"

 then 
                     "m = (x - y)*(x + y)"

. So if 
                          "1 < x - y"

 we have a non-trivial factorization of m. If m is a product of just two primes then one will be 
                            "x - y"

 and the other will be x + y.  This is the basis of the Fermat factorization method. Note that  
                        "m = x^2 - y^2"

 is equivalent to 
                        "x^2 - m = y^2"

.  So if for some number x we have type(sqrt(x^2 - m),integer) = true then we can set y = sqrt(x^2 - m) and using the above idea we may factor m.  Fermat's method is to take x = 
                        "ceil(sqrt(m))"

 and keep incrementing x by 1 till an x is found such that 
                           "x^2 - m"

 is a square.   [Note that you may need to use ceil(evalf(sqrt(m),200)).] 

If the number m is a product of two primes that are relatively close together then this method will factor m quickly. If you do not find the factors in just a few seconds, you probably have a bug in your program. For some reason Maple does not have this method builtin. 

[N.B. This exercise shows that if care is not taken in choosing the primes p and q used in the RSA system, than the system can be easily broken

Alice sent a message to Bob using the RSA system. The public key used to encode the message was 
[m,e] = [187454749788911503119994043213682616233000961, 23073697474256143563]. 

The enciphered message was
  [99763088506717050716379498520357841718875740, 120165413768425112311337085191282286020515442, 130554852152844850464253187688342192964186591, 30831038803473309186944039044693095063005267, 43791870977158745785820581340119798552660187, 69969965042359538654291028155927734707714502, 33869965925699990903801791477929460546990002, 33844215332363231914638130818590411552396086, 61034870951899306892357141802983347226683758, 152680702248950827958295294572717135277913525, 109591116592896191360922558098912453060002989, 155489947789995346755966254731407091292570943, 30508537854841352888337386800131678916944679, 45657204694505066046067342403378233363404150, 122580283375455701674982424288406984864898071, 63190645718352016104824679398188140019415424, 17253386795565064409576412640690151622064852, 76432681365602448262013510061731737725777808, 4419855123469897669096067167819380964828479, 87913005918401414546998775379426321338719933, 155434516526735215629181544556450169276890711, 74695381359901114443168866905755723039288800, 246854149837977729662442017899408430336317, 121766227774612543867361050340041336298158533, 166784352080096633524033563364825999607054575, 72405993088552642775735865911360107823805504]

Decipher this message. 

[Hint: Use of ifactor with the option 'pollard' may help. When you are finished with the decryption, you should use colons where possible to suppress unnecessary output. ]
 

Find, using Maple, a triple [m,e,d] where m is the product of two 200 decimal digit primes, e is a random prime number of 100 decimal digits and d is the inverse of e modulo 
                            "phi(m)"

. Use this to encipher some message of your choice of at least 6 lines length using the RSA system. You may copy and paste text from some website. Then, decipher the enciphered text to obtain the original message.