Download QuestionFile3Cryptography

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CS 413, Question File 3, Cryptography
Chapter 12 topics: Prime numbers, Euler's totient, RSA encryption
1. Define Euler’s totient function in words:
Φ(n) means:
2. If n is prime, give the value for Euler’s function:
Φ(n) =
3. Assuming n = pq where p and q are prime, give the value of Euler’s function:
Φ(n) =
4. If a and n are relatively prime, what does Euler’s theorem tell you about this
expression?
aΦ(n) ≡n
5. Based on Euler’s theorem, for a and n relatively prime, how does this expression
evaluate?
aΦ(n)+1 ≡n
6. RSA encryption involves, among other things, the plaintext M, ciphertext C, and
parameters p, q, e, d, and n. Write the formula for encrypting M:
C=
7. RSA encryption involves, among other things, the plaintext M, ciphertext C, and
parameters p, q, e, d, and n. Write the formula for finding M after encryption into C:
M=
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8. What is the defining mathematical characteristic of the parameter n in RSA
encryption?
9. Given p and q, and thus the product (p – 1)(q – 1), how should the parameter d be
chosen in RSA encryption?
10. What values or parameters are needed to encrypt in RSA, in other words, what is the
publicly distributed key?
11. What values or parameters are nominally identified as the private key in RSA? What
other parameters have to be kept secret?
12. By the way parameters are chosen in the RSA scheme, how does this evaluate?
ed ≡ (p-1)(q-1)
13. Let d and e have the characteristics defined in RSA encryption. Write an equation
(not a modular equivalence) for the value that the product e*d can take on using Φ(n), a
constant k, and any other items that might be needed in the expression.
14. Suppose some plaintext message M has been converted to a numeric form with a
decimal value less than n. If M and n are relatively prime, how do you expect this to
evaluate?
MΦ(n)+1 ≡n
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15. Under RSA encryption let p = 3, q = 11, and d = 13. Give the value for e.
16. Suppose the English alphabet is matched as usual with values 0-25. Using the
parameters of the previous problem, show the expressions for decimal integers you would
get if you were encrypting the following letters one at a time.
C=
A=
B=
17. Suppose that p = 5, q = 7, and d = 11. Assume the English alphabet is matched with
the decimal digits 0-25 as usual. A sequence of decimal digits which are the result of
RSA encryption is shown. Give the message that they decrypt to. In other words, find
the plaintext.
The sequence of decimal digits: 1, 0, 2, 2
18. Name three mathematical problems that are underlying components of RSA. Make
sure your answer includes the fundamental problem which renders RSA secure, and
identify it as such.
19. Suppose that you had computed in advance all prime numbers up through those with
a length of 200 decimal digits. Assume that you have sufficient secondary storage for
them and that you can discount paging costs because you can prefetch the primes in order
as needed. Then for some n <= 400 decimal digits in length, what would a simple brute
force attack on an RSA encryption scheme consist of?
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Possible Answers:
1. The number of integers i, 0 < i < n, such that i is relatively prime to n (gcd(i, n) = 1).
2. n – 1
3. (p – 1)(q – 1)
4. 1
5. a
6. Me mod n
7. Cd mod n
8. It is the product of just two primes, p and q.
9. It is relatively prime to (p – 1)(q – 1).
10. e and n.
11. Nominally: d and n. Others: p and q.
12. 1
13. ed = kΦ(n) + 1
14. M
15. 17
16. 29, 0, 1
17. BASS
18. Finding primes, finding inverses in modular fields, and factoring. Factoring is the
killer.
19. Divide n by each of the primes to see if it goes in evenly. If it does, and this is RSA
encryption, then the quotient (the other factor) had better also be in the list of primes.