Quiz Ch2

An Individual has a public key of N = 187(17 x 11) and e = 7

What is the private key d using RSA encryption?

A. 23

B. 22

C. 140

D. 42

A

phi = (p-1)(q-1) = (16 x 10) = 160

Y = (e x A multiple choice value) = 23 x 7

Y / phi = 161/ 160

=> 1 remainder 1

1 is the remainder and we know we've found our answer

https://www.lri.fr/~fmartignon/documenti/systemesecurite/6-PublicKey.pdf

What is 28^27 mod 76?

A. 5

B. 72

C. 27

D. 20

D

Step 1: Break down 27 into it's binary equivalent

27 = 11011 = 16 + 8 + 2 + 1

This also tells us that there will be 5 repetitions, because there are 5 digits needed to represent 27 in binary

Repetition 1: 28 mod 76 = 28 (2^0)

Repetition 2: 28^2 mod 76 = 784 mod 76 = 24 (2^1)

Repetition 3: 24^2 mod 76 = 576 mod 76 = 44 (2^2)

Repetition 4: 44^2 mod 76 = 1936 mod 76 = 36 (2^4)

Repetition 5 = 36^2 mod 76 = 1296 mod 76 = 4 (2^8)

Everywhere there is a 1 represented in binary corresponds with a value that we can use

(28 x 24 x 36 x 4) = 96768

=> 96768 mod 76 = 20

https://mathworld.wolfram.com/SuccessiveSquareMethod.html

Also verified here:

https://www.mathcelebrity.com/modexp.php?num=28%5E27+mod+76&pl=Successive+Squaring

You are give the following encryption method to shift characters in which A = 14, B = 15, C = 16, Z = 39, ...

How would one encode "Enter Password"?

A. 1827331831 2914323236283117

B. 1827330131 1514323219283117

C. 1827331835 2914323236282217

D. 1827241831 2914353236283117

A

It's ROT-13, take the position of the letter (starting with 1 and add 13 to it)

18 = E

27 = N

33 = T

18 = E

31 = R

29 = P

14 = A

32 = S

32 = S

36 = W

28 = O

31 = R

17 = D

What are the number of digits required to represent 1446 in base 7?

A. 6

B. 7

C. 4

D. 5

C

The number of digits to represent a number (n) in base (b) is determined by

Ceil(log(n + 1) / log(b))

log(1446 + 1) / log(7) = 3.7397655

rounding up: 3.7397655 gets us 4

https://www.geeksforgeeks.org/given-number-n-decimal-base-find-number-digits-base-base-b/

What is the sequence of remainders using Euclid's algorithm for gcd(106, 16)?

A. 10, 6, 4, 2 , 0

B. 12, 8, 4, 2, 0

C. 16, 14, 7, 1, 3

D. 16, 4, 2, 1

A

106 / 16 = 6 r10

16 / 10 = 1 r6

10 / 6 = 1 r4

6 / 4 = 1 r2

4 / 2 = 2 r0

https://scienceland.info/en/algebra8/euclid-algorithm

Alice wants to send a message to Bob. Alice chooses p = 11 and q = 3, e = 3.

If Bob gets the ciphertext 5 what is the plaintext message m?

A. 14

B. 5

C. 22

D. 33

A

m = (c^d) mod n

n = p x q

phi = (p - 1) x (q - 1)

(e x d) mod phi = 1 This is how we find d, because we know e and we know phi at this point.

n = 33

phi = 2 x 10 = 20

(3 x d) mod 20 = 1, the only thing that will get us a remainder of 1 would be 21 mod 20, therefore d = 7

Now we can use the equation:

m = (c^d) mod n

m = (5^7) mod 33

=> 78125 mod 33 = 14

Therefore A. 14 is our answer

https://www.cemc.uwaterloo.ca/resources/real-world/RSA.pdf

Alice wants to send a message to Bob. Alice chooses p = 11 and q = 3, e = 3. If the message(m) she wants to send is 14 what is the ciphertext C?

A. 22

B. 5

C. 9

D. 33

B.

C = (m^e) mod n

m = 14

e = 3

n = 33 (p x q)

14^3 = 2744

2744 mod 33 = 5

https://www.cemc.uwaterloo.ca/resources/real-world/RSA.pdf

What is A14(sub 16) in decimal?

A. 2580

B. 1243

C. 4869

D. 834

A

(10 x 16^2) + (1 x 16^1) + (4 x 16^0) = 2580

Verify here:

https://www.rapidtables.com/convert/number/hex-to-decimal.html

The binary representation of 42 is 101010. Which expression is equivalent to b^42

A. (b^2^8) x (b^2^4) x (b^2^0)

B. (b^2^16) x (b^2^4) x (b^2^0)

C. (b^2^5) x (b^2^3) x (b^2^1)

D. (b^2^4) x (b^2^3) x (b^2^0)

C

What is the multiplicative inverse of 8 Z(sub 15)?

A. 16

B. 14

C. 4

D. 2

D

2 x 8 = 16 mod 15 = 1 r1

Remainder 1 means we've found our answer

If today is Sunday, what day will it be in 15 days? 26 days? 234 days? 1000?

We need to find the remainder of 15, 26, 234 and 1000 when divided by 7. As we don't need the quotient we don't need to do the division. We can find the remainders writing the numbers as (multiples of 7) plus smaller numbers:

In 15=14+1 days it will be Monday.

In 26=21+5 days it will be Friday.

In 234=210+21+3 days it will be Wednesday.

In 1000=700+280+14+6 days it will be Saturday.

Source:

http://190.169.94.9/acm/main/entrenamiento/material/ModularArithmetic-Worksheets-Answers.pdf

Is the following an accurate modular congruence?

14 x 15 x 16 ≡ 6 (mod 3)

When we replace the numbers by their remainders modulo 3 we get:

2 x 0 x 1 ≡ 0 (mod 3) and this is true.

Source:

http://190.169.94.9/acm/main/entrenamiento/material/ModularArithmetic-Worksheets-Answers.pdf

Compute:

[(13 mod 6) + (53 mod 6)] mod 6

[(1) + (5)] mod 6 = [0] mod 6 = 0

Source:

http://190.169.94.9/acm/main/entrenamiento/material/ModularArithmetic-Worksheets-Answers.pdf

What is the ones digit of 3 to the 79 power?

This is a cycle problem:

Power = result / subsequent 1's digit

3^1 = 3 / 3

3^2 = 9 / 9

3^3 = 27 / 7

3^4 = 81/ 1

3^5 = 243/ 3

3^6 = 729/ 7

So we can see that this is a 4 cycle, therefore:

79 / 4 = 19 r3

So the cycle in the problem repeats 19 times, and then continues 3 numbers into the cycle.

So if we count out 3 numbers into the 4 cycle listed out above we land on 3^3 which has an associated 1's digit of 7, which is our answer

http://mathcentral.uregina.ca/QQ/database/QQ.09.05/candice1.html

What is the multiplicative inverse of 14 in Z(sub33)

26

26 * 14 = 364

364 mod 33 = 1

https://planetcalc.com/3311/