Question 2 Load Alice's public and private key data into the memory The values of R , R p , R q are all defined R has variable x in Sage, R p has variable y , and R q has variable z You can send a polynomial from R like Alice's f ( x ) , given by f in the data file, to its version in R q via the command R q ( f ) Verify that Alice's public key is correctly set up , that is , that f ( x ) h ( x ) g ( x ) in R q by running R q ( f ) h in Sage, and seeing that you get the same value as R q ( g ) Notice how your answers will have the z variable, to indicate that Sage is thinking of them as R q polynomials, whose coefficients are modulo q 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

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Question: Question 2 . Load Alice's public and private key data into the memory. The values of R , R p , R q are all

Question

2 .

Load Alice's public and private key data into the memory. The

values of

R, R p, R q

are all defined;

R

has variable

x

in Sage,

R p

has variable

y,

and

R q

has variable

z .

You can send a polynomial from R like Alice's

f (x),

given by

f

in the data file, to its version in

R_{q}

via the command

R q (f) .

Verify

that Alice's public key is correctly set up

,

that is

,

that

f (x) h (x) = g (x)

R_{q}

by running

R q (f) * * h

in Sage, and seeing that you get the same value as

R q (g) .

Notice how your answers will have the

z

variable, to indicate that Sage

is thinking of them as

R_{q}

polynomials, whose coefficients are modulo

q .

####################### 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 *

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