Chapter 5
Assume that we choose primes p = 29, q = 41, and n = 47. Apply the RSA cryptosystem guidelines to compute: z = Φ = s =
1189, 1120, 143
Assume that we choose primes p = 23, q = 59, and n = 355. Apply the RSA cryptosystem guidelines to compute: z = Φ = s =
1357, 1276, 895
Find gcd( 8232 , 8820 )
588 (with margin: 0)
Encrypt the message ARRIVING_TODAY using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: LTEBJZSQYD_VMWOIRUXCHKPFANG (use only capital letters and use _ for space)
TXXDPDOQLHIJTN
In the RSA public-key cryptosystem, which of the following information would allow an unauthorized person to decrypt a message?
Φ,p,s,q
A prime number is a number greater than ___ whose only positive divisors are _____
1, 1 and itself
To emphasize that the number 101101 should be regarded as a binary number, we will write
101101(/2)
Find gcd( 1650 , 2420 )
110 (with margin: 0)
Conver 2033(/10) in hexadecimal form.
127,1,127,7,15,7,0,7,7F1
Assume that we choose primes p = 19, q = 67, and n = 541. Apply the RSA cryptosystem guidelines to compute: z = Φ = s =
1273, 1188, 325
Assume that we choose primes p = 29, q = 53, and n = 233. Apply the RSA cryptosystem guidelines to compute: z = Φ = s =
1537, 1456, 25
Find gcd( 7800 , 6084 )
156 (with margin: 0)
Conver 2845(/10) in hexadecimal form.
177,13,177,11,1,11,0,11,B1D
Find lcm( 13860 , 12936 )
194,040 (with margin: 0)
Conver 3141(/10) in hexadecimal form.
196,5,196,12,4,12,0,12,C45
In the binary number 101010, the underlined 1 is in the
2's place
Conver 3478(/10) in hexadecimal form.
217,6,217,13,9,13,0,13,D96
The message b=688 has been encrypted using an RSA cryptosystem with public key (z,n)=(989, 41). Try to factor z, to determine p, q (p<q), �, and s, and use them to decrypt the message. p= q= Φ= s= decrypted message:
23,43,924,293,516
Encrypt 441 using an RSA cryptosystem with public key (1739, 541).
256
In the RSA public-key cryptosystem, if p=5 and q=11, which of the following would be acceptable values for n?
27,21,13
The message b=772 has been encrypted using an RSA cryptosystem with public key (z,n)=(1073, 53). Try to factor z, to determine p, q (p<q), �, and s, and use them to decrypt the message. p= q= Φ= s= decrypted message:
29, 37, 1008, 989, 607
Find lcm( 7800 , 6084 )
304,200 (with margin: 0)
Find lcm( 1650 , 2420 )
36,300 (with margin: 0)
The message b=261 has been encrypted using an RSA cryptosystem with public key (z,n)=(1517, 113). Try to factor z, to determine p, q (p<q), �, and s, and use them to decrypt the message. p= q= Φ= s= decrypted message:
37,41,1440, 497, 801
The message b=1495 has been encrypted using an RSA cryptosystem with public key (z,n)=(173, 95). Try to factor z, to determine p, q (p<q), Φ, and s, and use them to decrypt the message. p= q= Φ= s= decrypted message:
37,47,1656,767,1425
In the binary number 101000, the underlined 1 is in the
4's place
The message b=1594 has been encrypted using an RSA cryptosystem with public key (z,n)=(1763, 127). Try to factor z, to determine p, q (p<q), Φ, and s, and use them to decrypt the message. p= q= Φ= s= decrypted message:
41, 43, 1680, 463, 777
Find gcd(4158, 15288)
42 (with margin: 0)
Conver 85(/10) in binary form.
42,1,42,21,0,21,10,1,10,5,0,5,2,1,2,1,0,1,0,1 1010101
Conver 88(/10) in binary form
44,0,44,22,0,22,11,0,11,5,1,5,2,1,2,1,0,1,0,1, 1011000
Conver 92(/10) in binary form.
46,0,46,21,0,23,11,1,11,5,1,5,2,1,2,1,0,1,0,1, 10111000
Conver 93(/10) in binary form
46,1,46,23,0,23,11,1,11,5,1,5,2,1,2,1,0,1,0,1, 1011101
Decrypt the message OACFQMKTQMBOGH using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: MOARIQPZTGX_VEHCYWBJKDFSUNL (use only capital letters and use _ for space)
ABOVE_THE_RAIN
In the hexadecimal system, the number 11 is represented by
B
Encrypt the message SEE_YOU_TOMORROW using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: LTEBJZSQYD_VMWOIRUXCHKPFANG (use only capital letters and use _ for space)
CZZLNIKLHIWIXXIF
In the hexadecimal system, the number 13 is represented by
D
In the hexadecimal system, the number 14 is represented by
E
Encrypt the message TURN_TO_THE_LEFT using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: FNCWKITZLYHUMSRP_XDQEABJGOV (use only capital letters and use _ for space)
EADRFEPFELIFMITE
Decrypt the message LBDZAKWBKWLAKSAPW using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: KMQFEAPILHRGSUCBYNTZWDJXO_V (use only capital letters and use _ for space)
HOUSE_TO_THE_LEFT
Encrypt the message ARRIVING_TODAY using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: ORWFKUZHEPLTVXDQ_YJGBIMSNCA (use only capital letters and use _ for space)
RJJPMPDHOBQKRC
Decrypt the message XUEEUESQKUIJHK using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: QVLIAHWSFUBJZCE_NDMOKYTXPRG (use only capital letters and use _ for space)
WINNING_TICKET
If integers in the range 0 to m,m⩾8, not both zero, are input to the Euclidean algorithm, then the required number of modulus operations is
at most log(/(3/2) 2m/3
Suppose that the pair a,b,a>b, requires n⩾1 modulus operations when inputto the Euclidean algorithm. If {f(/n)} denotes the Fibonacci sequence, then
a⩾f(/n+2 and b⩾f(/n+1.
A binary digit is also referred as a
bit
A positive integer number greater than or equal to 2 that is not prime is called
composite
In the late 1900s, Number Theory became extremely useful in
cryptosystems
In a private-key cryptosystem, the sender __ the key and the reciever ____ the key.
does not disclose, does not disclose
In the RSa public-key cryptosystem
each participant discloses his/her encruption key
In a cryptosystem, the sender __ the message, and the reciever ___ the message
encrypts and decrypts
The fibonacci sequence {f(/n)}is defined by
f(/1) = 1 f(/2)=1 f(n) = f_{n-1} + f_{n-2}
a problem for which a polynomial time alogrithm exists is said to be solvable
false
The Euclidean Algorithm is an efficient algorithm for
finding the gcd of two integers numbers
The Euclidean Algorithm is based on the fact that, if r= a mod b, then
gcd(a,b)=gcd(b,r)
Let a, b, and c be integers. Show that, if a|b and b|c, then a|c.
given given 1 definition of a|b 2 definition of b|c c=(aq)r 5 associative property a|c 6 and efinition of a|c
Let m, n, and d be integers. Show that, if d|m and d|n, then d|m-n.
given given m-dq for some q 1 n=dr for some r 2 3 4 5 definition of d|m-n
Let m, n, h, and k be integers. Show that, if h|m and k|n, then hk|mn.
given given m=hq for some q 1 n-kr for some r 2 3 4 5 commutativity of the product 6 definition of hk|mn
Let m, n, and d be integers. Show that, if d|m, then d|mn.
given m=dq for some q 1 mn=dqn for some q 2 3 definiton of d|mm
gcd(m,n) * lcm(m,n) is equal to
mn
In the RSA public-key cryptosystem, which of the following information will be shared with the public?
n ,z
Let n and Ø be positive integers. The inverse of n Ø mod is s such that
ns mod Ø = 1
If the prime factorization of m and n are respectively m=p(a1/1),p(a2/2)...p(ak/k) n=p(b1/1),p(b2/2)...p(bk/k) then lcm(m,n) is equal to
p1(max(a1,b1) p2max(a2,b2)...pkmax(ak,bk)
If the prime factorization of m and n are respectively m=p(a1/1),p(a2/2)...p(ak/k) n=p(b1/1),p(b2/2)...p(bk/k) then gcd(m,n) is equal to
p1(min(a1,b1) p2(min(a2,b2)...pk(min(ak,bk)
Cryptology is the study of systems, called cryptosystems, for __ communication
secure
In the RSA public-key cryptosystem, z is
the product of two large prime numbers
RSA public-key cryptosystem is secure because
there is no efficient algorithm to factor z
NP denotes the class of problems for which a solution can be guessed and verified in polynomial time
true
