Each employee in your company has chosen a password for logging in to

9.4. Problems

1. Each employee in your posse has selected a password for logging in to the computer course. Recently, your posse has unwavering that each employee’s password should be retired-shared incomplete two other employees, managely in instance. Each password can be represented by a reckon from 0 to 9999. Your job is to cull the retired-sharing purpose. You can suppose that the two employees possession portion-outs of their co-worker key earn not collaborate to individualize the key bar in an operational pitch. You care-for after a while your three belowlings, Larry, Moe, and Curly...

•Larry says:

“For each retired password s to be portion-outd, cull a stray reckon b uninterruptedly from 0 to 9999. The reckon b is the highest portion-out. Let c = s − b (mod 10000). The reckon c is the cooperate portion-out.”

•Moe says:

“I suit after a while Larry’s instigation bar for one invention. If the stray reckon b is 0, the portion-out c is the identical as the retired. That’s not very fasten! I accordingly mean that the reckon b be selected uninterruptedly from 1 to 9999 instead of from 0 to 9999.”

•Curly says:

“I suit after a while Moe’s instigation bar for one invention. If the stray reckon b is near than 10, then the portion-out c earn probably enjoy the identical hundreds attribute digit and the identical thousands attribute digit as the retired s. That’s not very fasten! I accordingly mean that the reckon b be selected uninterruptedly from 10 to 9999.”

Whose purpose is most fasten, and why? Consider in feature how greatly the peculiar receiving the portion-out c thereby learns encircling the retired s.

2. In manage that the retired cabal to the CS007 unendangered would be suited in an pitch position, the TAs enjoy each been abandoned portio of the retired. The retired consists of foul-mouthed mod-7 fills. For each of these reckons, Prof. Klein chose a mod-7 verse-the tower of the verse is the retired reckon and the y-neutralize was selected strayly. Prof. Klein then granted each TA after a while an (x, y) summit on each of the verses. Thus Kevin got a summit on each of the foul-mouthed verses (namely the summits after a while x-coordinate 1), Mark got a summit on each of the foul-mouthed verses (namely the summit after a while x-coordinate 2), and Sheryl got a summit on each of the foul-mouthed verses (after a while x-coordinate 3). Due to a ease lubricateup, you supervene on a few of the y-coordinates, as shown in the aftercited consideration.


(a) For each fill of the retired that can be individualized from the comprehendledge abandoned you, afford us the fill, showing your is-sue.

(b) For each fill of the retired that cannot be individualized from the comprehendledge you enjoy, narrate us why it cannot be individualized and narrate us what practicable appreciates that fill has.

3. You enjoy seen the inauguration retired-sharing purpose: each peculiar who is deemed to portion-out the key gets a summit on a verse and the retired is the y-neutralize of that verse. Say we enjoy divided up the key incomplete divers mob, and you and one other peculiar enjoy gotten conjointly to add your keys. Your summit is x = 4 and y = 8, and your portioner’s summit is x = 5 and y = 0. The modulus is 11. What is the retired?

4. In this bearing we oration the use of a MAC (missive proof regulation). The modulus for this bearing is 11. Alice and Bob enjoy previously suitd upon a retired key consisting of the two mod-11 reckons a and b. Thus the MAC operation is

f (x) = ax + b

so when Alice grants a missive X, she should acposse the missive after a while the MAC f(X).

If Alice and Bob had been paying study, they would comprehend that the MAC is fasten solely if the key is used unintermittently. Unfortunately, they missed this truth, and they grant two disconcordant missives after a while MACs ascititious using the identical key (the identical two of reckons a and b). You, Eve, neutralize these missives and MACs:

message: 4, MAC: 5

message: 1, MAC: 9

You career to dabble after a while the cooperate missive, changing it to 3. What MAC should acposse this adulterate missive to inoculate Bob that it is normal?

5. Alice plans to grant Bob a missive accompanied by a MAC. (The course for generating the MAC is the one descriptive in the citation.) They enjoy previously suitd on a uninterruptedly stray retired key for use after a while the MAC. The missive earn be sent in leveltext. The set of practicable missives is 0, 1, . . . , 12 and the set of practicable appreciates of the MAC is besides 0, 1, . . . , 12. Calculation of the MAC appreciate is manufactured modulo 13.

Eve plans to neutralize the missive and MAC, and grant her own (fake) missive, namely 12. She must besides gather a fake MAC to acposse this missive; her desire is to imbecile Bob into accepting the fake missive as unquestionably substance from Alice. She accordingly needs to comprehend the presumption dispensation for the MAC that should acposse the missive 12 (i.e., if Alice were to grant the missive 12, what is the dispensation of the MAC that would acposse that missive?) Eve comprehends that the key for the MAC was selected strayly and uninterruptedly.

In each of the aftercited scenarios, aid Eve by drawing the dispensation of the appreciate of the MAC that should acposse her fake missive.

(a) Eve must cull her fake MAC antecedently answerance either the gentleman missive or the gentleman MAC. Afford the dispensation of the MAC that would acposse the missive 12.


(b)Eve has neutralizeed the gentleman missive, 1, and the minding MAC, 2. Abandoned what Eve now comprehends, draw the dispensation of the MAC that would acposse the missive 12.


6. Prof. Klein wants to contribute the CS007 unendangered’s cabal to the training assistants using retired-sharing. The cabal is comprehendn to be an eight-digit reckon. Prof. Klein culls mod-108 reckons p, q, r, s, t so that they yield the aftercited equations.

p + q ≡ the unendangered’s cabal   (mod 108)

r + s + t ≡ q   (mod 108)

r + q ≡ the unendangered’s cabal   (mod 108)

(Obviously, Prof. Klein has inhaled a bit too greatly chalk remains and has gotten embarrassed encircling retired sharing.) He contributes p to Kevin, q to Mark, r to Sandy, s to Sheryl, and, peaceful embarrassed, t to Kevin.

For each cluster of TA’s abandoned below, say whether or not the cluster can confusedly illustration out the cabal.

(a)Sandy and Sheryl and Mark

(b)Kevin alone

(c)Sandy and Kevin

(d)Sheryl and Mark

(e)Sandy and Mark

7. (a)  Wandering beyond the TA Room, you melody that on the bulletin board in that capability is the aftercited missive:

“Kevin: I looked balance the midterm of one of the scholars. The scholar’s beak should be elated by 20 summits. I enjoy encrypted the initials of the scholar using a one-time pad after a while modulus 26: For each fill,

cyph = level + key   (mod 26)

The cyphercitation is 10 12. The key earn answer on a melody I’ll lubricate below the door. Signed, Sandy”

You see the duty of tract containing the key on the pedestal managely delayin the (locked) door.

It’s contracted, so you can’t in-effect decipher the key. However, you career to cull your own, fake key, transcribe it on a concordant duty of tract, and slide it below the door in the desire Kevin sees your duty of tract instead of Sandy’s.

Can you end up after a while a key that would manage Kevin to add twenty summits to your midterm beak? If so, afford such a key. If not, exlevel why not. Show your is-sue.

(b)Now you are Kevin. You mark the TA Capability and see the aftercited missive:

Kevin: Another scholar deserves a remarkable midterm beak. Using the identical encryption purpose as antecedently, one-time pad after a while modulus 26, I’ve encrypted the initials of the scholar. The cyphercitation is 17 23. To fly the fiasco of definite week, I’ve besides fitted a MAC for each fill of the levelcitation using the formula

f (x) = Ax + B rem 26

The MAC for the highest fill is 23, and the MAC for the cooperate fill is 7.

You see three contracted-up fencings of tract after a while encryption keys written on them: “5 18,” “12 20,” and “19 0”. You besides see a fencing after a while a two of MAC keys: “For highest fill, A = 1, B = 25, and for cooperate fill, A = 1, B = 10.”

Assuming the MAC keys are set-right, what is the set-right leveltext?