Question 3 Bob has sent Alice an encrypted message e ( x ) , given as e 1 in the data file Load this value into memory, and decrypt it using Alice's private key You can use the centerlift function to compute the centered life from R p or R q to R , so for example, once you compute R q ( f ) e 1 in R q , you can run a centerlift ( R q ( f ) e 1 ) to get a ( x ) in its centered lift form in R Notice this will be a polynomial in x Remember, you can find the inverse of a polynomial f ( x ) in R p or R q via R p ( f ) 1 or R q ( f ) 1 Don't forget that when you find the centered lift, your polynomial is now in R , so you will need to convert it to Rp via Rp ( a ) Finally, you can use the decode function to decode your m ( x ) , FUNCTIONS You can safely run this entire block at any point until you see the next block begin below def T ( d 1 , d 2 , N ) Returns a random ternary polynomial in ZZ z with d 1 coefficients that equal 1 d 2 coefficients that equal 1 if d 1 d 2 N raise ( ValueError , d 1 d 2 cannot exceed N ) R PolynomialRing ( ZZ ) S Set ( range ( N 1 ) ) pos S subsets ( size d 1 ) random element ( ) S 2 S pos neg S 2 subsets ( size d 2 ) random element ( ) v 0 for i in range ( N 1 ) for i in pos v i 1 for i in neg v i 1 return R ( v ) def centerlift ( f ) R PolynomialRing ( ZZ ) P f parent ( ) base ring ( ) cardinality ( ) return R ( mod ( t , P ) lift centered ( ) for t in f ) def encode ( s ) Encode the message ASCII string s as a polynomial m ( x ) in R p Uses environment variables p and N s str ( s ) k N log ( p , 1 2 8 ) if len ( s ) k raise ( ValueError , String too long Max length in R p is str ( floor ( k ) ) ) t sum ( ord ( s i ) 1 2 8 i for i in range ( len ( s ) ) ) return Rp ( t digits ( base p ) ) def decode ( m ) Decode the polynomial m ( x ) in R p mc list ( m ) Remember, mc i belongs to GF ( p ) Lift to integers c sum ( Integer ( mc i ) p i for i in range ( len ( mc ) ) ) v ' ' join ( chr ( x ) for x in Integer ( c ) digits ( base 1 2 8 ) ) return v def circulantmatrix ( a ) Returns the matrix whose ROWS are the rotations of the vector a ( a 0 a 1 a n 1 ) ( a n 1 a 0 a n 2 ) ( ) ( a 1 a 2 a 0 ) n len ( a ) R range ( n ) return matrix ( a ( j k ) n for j in R for k in R ) def NTRUmatrix ( h ) Returns the NTRU lattice matrix associated to the public key ( N , p , q , d ) and h ( y ) in R q , in row form This matrix is , in ROW form, the 2 N x 2 N block matrix ( I H ) ( 0 qI ) where H is the circulant matrix generated by the coefficients of h as a vector hv list ( centerlift ( h ) ) H circulantmatrix ( hv ) I identity matrix ( ZZ , N ) O zero matrix ( ZZ , N ) return block matrix ( I , H , O , q I ) End of functions block PROBLEMS 2 onwards Alice's public key p 7 2 3 8 2 9 q 4 1 4 1 6 5 4 1 3 N 1 5 1 d 3 0 R PolynomialRing ( ZZ ) Sp PolynomialRing ( GF ( p ) ) Rp Sp quotient ( Y N 1 ) Sq PolynomialRing ( GF ( q ) ) Rq Sq quotient ( Z N 1 ) h 2 9 2 1 5 4 1 8 9 z 1 5 0 3 0 1 4 7 3 7 5 1 z 1 4 9 7 9 9 2 2 7 9 z 1 4 8 2 5 6 2 7 4 5 7 7 z 1 4 7 1 3 8 2 9 1 7 9 2 z 1 4 6 2 1 3 2 4 4 8 0 0 z 1 4 5 6 6 0 1 8 6 6 4 z 1 4 4 2 1 6 5 4 1 6 5 3 z 1 4 3 3 6 5 2 1 9 3 9 5 z 1 4 2 8 2 8 5 7 9 5 0 z 1 4 1 1 2 8 6 6 0 5 3 8 z 1 4 0 2 9 3 4 2 0 8 7 6 z 1 3 9 7 3 2 0 3 0 4 3 z 1 3 8 2 6 0 7 9 9 1 8 9 z 1 3 7 3 5 9 9 0 7 2 4 6 z 1 3 6 3 4 9 4 1 7 3 0 4 z 1 3 5 1 0 2 6 4 2 6 0 8 z 1 3 4 1 1 8 6 0 3 1 2 5 z 1 3 3 4 0 9 6 6 9 2 5 4 z 1 3 2 9 4 9 9 9 9 7 3 z 1 3 1 1 1 4 6 5 6 2 5 z 1 3 0 2 2 8 1 0 1 5 3 9 z 1 2 9 1 0 9 6 0 6 8 5 2 z 1 2 8 8 3 2 8 2 2 9 9 z 1 2 7 3 9 8 9 8 8 8 3 6 z 1 2 6 4 2 7 3 6 6 0 0 z 1 2 5 4 8 0 0 9 7 6 9 z 1 2 4 1 8 8 8 3 2 1 9 2 z 1 2 3 7 5 4 0 0 4 9 2 z 1 2 2 2 3 8 2 8 6 1 0 3 z 1 2 1 2 7 3 3 8 5 0 6 z 1 2 0 2 1 3 0 9 9 0 8 z 1 1 9 1 0 9 0 8 0 9 4 4 z 1 1 8 7 5 5 8 0 8 2 2 z 1 1 7 3 7 0 7 8 3 4 3 9 z 1 1 6 6 4 3 3 8 6 0 0 z 1 1 5 1 4 0 8 5 7 7 8 5 z 1 1 4 1 9 7 1 0 9 8 6 4 z 1 1 3 2 3 9 8 1 7 4 1 8 z 1 1 2

The Answer is in the image, click to view ...

Question: Question 3 . Bob has sent Alice an encrypted message e ( x ) , given as e 1 in the data file. Load this

Question

3 .

Bob has sent Alice an encrypted message

e (x),

given as e

1

the data file. Load this value into memory, and decrypt it using Alice's private

key. You can use the centerlift function to compute the centered life from

R_{p}

R_{q}

R,

so for example, once you compute

R q (f) * * e 1

R q,

you can

run

a =

centerlift

(R q (f) * * e 1)

to get

a (x)

in its centered lift form in

R .

Notice this will be a polynomial in

x .

Remember, you can find the inverse of

a polynomial

f (x)

R_{p}

R_{q}

via

R p (f)^{- 1}

R q (f)^{- 1} .

Don't forget

that when you find the centered lift, your polynomial is now in R

,

so you will

need to convert it to Rp via Rp

(

) .

Finally, you can use the decode function

to decode your

m (x) ., .

####################### FUNCTIONS #######################

### You can safely run this entire block at any point ###

### until you see the next block begin below. ###

def T

(

1,

2,

)

" " "

Returns a random ternary polynomial in ZZ

[

]

with:

1

coefficients that equal

+ 1

2

coefficients that equal

- 1

" " "

if d

1 +

2 >

raise

(

ValueError

, "

1 +

2

cannot exceed N

")

. =

PolynomialRing

(

)

=

Set

(

range

(

+ 1))

pos

=

.

subsets

(

size

=

1) .

random

_

element

()

2 =

-

pos

neg

=

2 .

subsets

(

size

=

2) .

random

_

element

()

= [0

for i in range

(

+ 1)]

for i in pos:

[

] = 1

for i in neg:

[

] = - 1

return R

(

)

def centerlift

(

)

. =

PolynomialRing

(

)

=

.

parent

() .

base

_

ring

() .

cardinality

()

return R

([

mod

(

,

) .

lift

_

centered

()

for t in f

])

def encode

(

)

" " "

Encode the message ASCII string s as a polynomial m

(

)

in R

_

.

Uses environment variables p and N

.

" " "

=

str

(

)

=

*

log

(

, 128)

if len

(

) > =

raise

(

ValueError

,

"String too long. Max length in R

_

p is

" +

str

(

floor

(

)))

=

sum

(

ord

(

[

]) * 128^

i for i in range

(

len

(

)))

return Rp

(

.

digits

(

base

=

))

def decode

(

)

" " "

Decode the polynomial m

(

)

in R

_

.

" " "

=

list

(

)

# Remember, mc

[

]

belongs to GF

(

) .

Lift to integers!

=

sum

(

Integer

(

[

]) *

^

i for i in range

(

len

(

)))

='' .

join

([

chr

(

)

for x in Integer

(

) .

digits

(

base

= 128)])

return v

def circulantmatrix

(

)

" " "

Returns the matrix whose ROWS are the rotations of the vector a:

(

[0]

[1] . . .

[

- 1])

(

[

- 1]

[0] . . .

[

- 2])

(. . . . . .)

(

[1]

[2] . . .

[0])

" " "

=

len

(

)

=

range

(

)

return matrix

([[

[(

-

) %

]

for j in R

]

for k in R

])

def NTRUmatrix

(

)

" " "

Returns the NTRU lattice matrix associated to the public key

(

,

,

,

)

and h

(

)

in R

_

,

in row form.

This matrix is

,

in ROW form, the

2

N x

2

N block matrix:

(

I H

)

(0

)

where H is the circulant matrix generated by the coefficients of h

as a vector.

" " "

=

list

(

centerlift

(

))

=

circulantmatrix

(

)

=

identity

_

matrix

(

,

)

=

zero

_

matrix

(

,

)

return block

_

matrix

([[

,

], [

,

*

]])

### End of functions block ###

###################################################

#################### PROBLEMS

2

onwards ####################

#################### Alice's public key ####################

= 723829

= 414165413

= 151

= 30

. =

PolynomialRing

(

)

. =

PolynomialRing

(

(

))

. =

.

quotient

(

^

- 1)

. =

PolynomialRing

(

(

))

. =

.

quotient

(

^

- 1)

= 292154189 *

^150 + 301473751 *

^149 + 7992279 *

^148 + 256274577 *

^147 + 138291792 *

^146 + 213244800 *

^145 + 66018664 *

^144 + 216541653 *

^143 + 365219395 *

^142 + 82857950 *

^141 + 128660538 *

^140 + 293420876 *

^139 + 73203043 *

^138 + 260799189 *

^137 + 359907246 *

^136 + 349417304 *

^135 + 102642608 *

^134 + 118603125 *

^133 + 409669254 *

^132 + 94999973 *

^131 + 11465625 *

^130 + 228101539 *

^129 + 109606852 *

^128 + 83282299 *

^127 + 398988836 *

^126 + 42736600 *

^125 + 48009769 *

^124 + 188832192 *

^123 + 75400492 *

^122 + 238286103 *

^121 + 27338506 *

^120 + 21309908 *

^119 + 109080944 *

^118 + 75580822 *

^117 + 370783439 *

^116 + 64338600 *

^115 + 140857785 *

^114 + 197109864 *

^113 + 239817418 *

^112 +<Step by Step Solution There are 3 Steps involved in it 1 Expert Approved Answer Step: 1 Unlock Question Has Been Solved by an Expert! Get step-by-step solutions from verified subject matter experts View Solution Step: 2 Unlock Step: 3 UnlockStudents Have Also Explored These Related Programming Questions! Q: Need Help... Instructions: This programming assignment is designed to develop your Scheme programming skills and to illustrate how functional programming can be used for security applications. In... Q: Suppose Alice wants to send some sensitive information (credit card numbers, SSNs, corporate secrets, health records, invasion orders to start a land war in Asia ) to Bob. To prevent eavesdroppers... Q: Need help on this MIPS assignment about subroutines. I'm stuck on how to allocate memory and checksums. Anyone could help me out I would appreciate. Lab Objective In this lab, you will learn how to... Q: Objectives 1. Working on a project that utilizes a broad spectrum of knowledge from this course 2. Using various forms of indirect addressing 3. Passing parameters on the stack 4. Extensive... Q: Need help on this MIPS assignment about subroutines. I'm stuck on how to allocate memory and check sums. Any one could help me out I would appreciate. Lab Objective In this lab, you will learn how to... Q: Developments in Technology Light is incident from air on the end face of a multimode optical fibre at angle of incidence  as shown below. n n 1 2  The refractive indices of the core and cladding are... Q: ITBP301 Security Principles and Practices Lab#2: ASYMMETRIC ENCRYPTION Objectives Giving you hands-on experience with RSA algorithm and the RSA key generation process. Overview: Public-Key... Q: QUESTION 1 Suppose Alice wants to communicate confidentially with Bob using asymmetric cryptography. Alice should encrypt the message to Bob using which of the following keys? Bob's private key... Q: I need help with this please. Essentially I need to complete each of the functions listed within the directions. The main functions are just for testing the listed functions. A sample output would be... Q: **Need Help In C Programming ASAP** Abstract In this programming assignment, you will implement a Fibonacci function that avoids repetitive computation by computing the sequence linearly from the... Q: Read the case Inventory Management for Pharmaceutical Industry and answer the following questions: 1. How well do you think the existing inventory control system works? What are the current annual... Q: Affirmative action has become a political, economic, and emotional issue. What are some of the reasonable arguments in favor of and opposed to affirmative action. Q: hich of the following internal controls provides reasonable assurance that tangible long - lived assets exist? a . Physical asset inventory is reconciled with the property ledger on a periodic basis.... Q: 3. Identify and describe the stages of group development in Bruce Tuckman's five-stage model. Provide an example of a situation where a group was stuck in one of the stages and how it could have been Previous Question Next QuestionRecommended Textbook More Books The History Of Visual Magic In Computers How Beautiful Images Are Made In Cad 3d Vr And Ar Authors: Jon Peddie 2013 Edition 1447149319, 978-1447149316 Ask a Question and Get Instant Help! Get AnswerStudy Help Open in AppServices Sitemap Fun Definitions Become Tutor Used Textbooks Study Help Categories Recent Questions Expert Questions Campus Wear Sell Your Books Company Info Security Copyrights Privacy Policy Terms & Conditions SolutionInn Fee Scholarship Online Quiz Give Feedback, Get Rewards Get In Touch About Us Contact Us Career Jobs FAQ Student Discount Campus Ambassador Secure Payment Download Our App © 2026 SolutionInn. All Rights Reservedwindow.addEventListener("load",function(){jQuery(document).ready(function(t){

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