Question 7 Bob's actual private key looked like f x 4 9 x 4 7 x 4 6 x 4 4 dots x 2 x 1 g x 5 0 x 4 9 x 4 8 x 4 7 dots x 3 x 2 x 1 Did Eve's key f 2 , g 2 give the same polynomials If not, why did they still work 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 3 8 4 3 2 9 4 8 2 z 1 1 1 3 2 9 5 3 0 9 5 7 z 1 1 0

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

Question: Question 7 . Bob's actual private key looked like: f = - x 4 9 - x 4 7 + x 4 6 - x

Question

7 .

Bob's actual private key looked like:

f = - x^{49} - x^{47} + x^{46} - x^{44} -

dots

+ x^{2} - x + 1

g = - x^{50} + x^{49} + x^{48} + x^{47} +

dots

+ x^{3} - x^{2} + x + 1

Did Eve's key f

2,

2

give the same polynomials? If not, why did they still

work?

\

####################### 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 + 384329482 *

^111 + 329530957 *

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