Selecting Class Polynomials for the Generation of Elliptic Curves

Elisavet Konstantinou

joint work with Aristides Kontogeorgis

Department of Information and Communication Systems Engineering

University of the Aegean

2

Why Elliptic Curves?

More Efficient (smaller parameters)

Faster

Less Power and Computational Consumption

Cheaper Hardware (Less Silicon Area, Less

Storage Memory)

3

Frequent Generation of ECsRequests different EC parameters

(due to security requirements, vendor preferences/policy etc.)

Frequent change of parameters calls for strict timing response constraints

4

Generation of ECs

The goal is to determine the following parameters of an EC

y2 = x3 + ax + b

The order p of the finite field Fp.

The order m of the elliptic curve.The coefficients a and b.

5

Generation of secure ECs

Cryptographic Strength suitable order m

Suitable order m = nq where q a prime > 2160

m p pk ≢ 1 (mod m) for all 1 k 20

The above conditions guarantee resistance to all known attacks

Sometimes, a prime m may be additionally required

6

Generation of ECs

Point Counting methods: Rather slow

(with )

ECs have to be tried before a prime order EC is found in Fp

Complex Multiplication (CM) method: Rather involved implementation, but more efficient

first the order is selected and then the EC is constructed

p

cp

log62.044.0 pc

7

Complex Multiplication method Input:a prime p

Class polynomial Hilbert polynomial

Transform the roots

Construct the EC

Determine D s.t. 4p=x2+Dy2 for x,y integers

EC order m=p+1 x

Is the order m suitable?

NO YES

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Class field polynomials

Class field polynomials: polynomials with integer coefficients whose roots (class invariants) generate the Hilbert class field of the imaginary quadratic field K = Q( ).

Drawback of Hilbert polynomials: large coefficients; time consuming construction; difficult to implement in devices of limited resources.

other class field polynomials: much smaller coefficients.

D

9

Class field polynomials

Alternative class field polynomials:

1) Weber polynomials

2) MD,l(x) polynomials

3) MD,p1,p2(x) polynomials or Double eta polynomials

4) Ramanujan polynomials TD(x)

All are associated with a modular polynomial Φ(x, j) that transforms a root x of these polynomials to a root j of the Hilbert polynomial.

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An example (D = 292)

W292(x) = x4 - 5x3 - 10x2 - 5x + 1

H292(x) = x4 - 2062877098042830460800 x3 - 93693622511929038759497066112000000x2 +

45521551386379385369629968384000000000x 380259461042512404779990642688000000000000

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Congruences for D

D ≢ 0 mod 3 D 0 mod 3

d = D/4

if D 0 mod 4

d = D

if D 3 mod 4

MD,l polynomials Ramanujan polynomials Double eta polynomials

D 0 mod l

Weber polynomials

1

2 or 6

3

5

7

d mod 8

1

2 or 6

3

5

7

d mod 8

1,121

p

D

p

DD 11 mod 24

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Hilbert polynomials

))(()( jxxH D

a

Db

2

satisfies the equation 02 cbxax

(primitive, reduced quadratic forms)

D [a, b, c] h

THEOREM:

A Hilbert polynomial with degree h, has exactly h roots modulo p if

and only if the equation 4p=x2+Dy2 has integer solutions.

13

Weber polynomials

l

D lgxxW ))(()(

a

Dbl

g is defined by the Weber functions f, f1 and f2

satisfies the equation 022 cbxax

[a, b, c]D h or 3h

(quadratic forms)

The degree of Weber polynomials is 3 times larger than thedegree of the corresponding Hilbert polynomials when D ≡ 3 mod 8.

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MD,l(x) polynomials

Q

QellD mxxM

))(()(,

A

DBQ 2

where 13,7,5,3l and e depends on l

satisfies the equation 02 CBxAx

(primitive, reduced quadratic forms)D [a, b, c] h[A, B, C]

2 transf.

divisible by l

each root RM is transformed to a Hilbert root

RH with a modular equation:

0),( HMl RR

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MD,p1,p2(x) polynomials

Q

QppppD mxxM

))(()( 2,12,1,

A

DBQ 2

where 21, pp primes and

satisfies the equation 02 CBxAx

(primitive, reduced quadratic forms)D [a, b, c] h[A, B, C]

2 transf.

each root RMd is transformed to a Hilbert root

RH with a modular equation (which has large coefficients and degree at least 2 in RH ):

0),(2,1 HMdpp RR

11

p

D

12

p

D

)1)(1(24 21 pp

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Ramanujan polynomials TD(x)

THEOREM:

The Ramanujan value tn is a class invariant for n 11 mod 24.

Its minimal polynomial is equal to:

))(()( txxTD

a

Db

2

satisfies the equation 02 cbxax and the construction

of the function t() is based on modular functions of level 72.

Precision Requirements

Bit precision for the construction of polynomials EQUAL to logarithmic height of the polynomials

17

011

1)( axaxaxaxg hh

hh

ihi

a2,...,0

logmax

Bit precision for the Hilbert polynomials:

],,[

1

2ln33)(Pr

CBA A

DDecH

Precision Requirements

“Efficiency” of a class invariant is measured by the asymptotic ratio of the logarithmic height of a root of the Hilbert polynomial to a root of the class invariant.

Asymptotically, one can estimate the ratio of the logarithmic height h(j(τ)) of the algebraic integer j(τ) to the logarithmic height h(f(τ)) of the algebraic integer f(τ). Namely,

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Precision Requirements

Let H(Pf) be the logarithmic height of the minimal polynomial of the algebraic integer f(τ) and H(Pj) the logarithmic height of the corresponding Hilbert polynomial. Then,

where m = 1 if f(τ) generates the Hilbert class field and

m = extension degree when f(τ) generates an algebraic extension of the Hilbert class field.

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m

fr

jf

jf

PH

PH

j

f

f

j )(

),(deg

),(deg

)(

)(

Precision Requirements

We can derive the precision requirements for the construction of every class polynomial by the equation

In all cases m = 1, except when D ≡ 3 mod 8 for Weber polynomials.

20

],,[

1

2ln)( CBA A

D

fr

m

Ramanujan polynomials

The modular equation for Ramanujan polynomials is:

Therefore, the value r(f) = 36. Also, since the degree of Ramanujan polynomials is equal to the degree of Hilbert polynomials, the value m = 1.

Theoretically, there is a limit for r(f) ≤ 96. The best known value is r(f) = 72 for Weber polynomials with D ≡ 7 mod 8.

21

0)276(),( 183612 HTTTHTT RRRRRR

Precision Estimates

22

Precision Estimates

23

Precision Estimates

24

Experiments

26

Construction of polynomials (bit prec.)

27

Construction of polynomials (bit prec.)

28

Experimental observations

The precision requirements for the construction of Ramanujan polynomials are on average 66%, 42%, 32% and 22% less than the precision requirements of MD,13(x), Weber, MD,5,7(x) and MD,3,13(x) respectively. The percentages are much larger when other MD,l(x)

and MD,p1,p2(x) polynomials are used.

The same ordering is true for the storage requirements of the polynomials with one exception: Weber polynomials.

13,7,5,13,3, DDD MWeberMMRamanujan

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Conclusions

Ramanujan polynomials clearly outweigh all previously used polynomials when D ≡ 3 mod 8 and they are by far the best choice in the generation of prime order ECs.

The congruence modulo 8 of the discriminant is crucial for the size of polynomials and this affects the efficiency of their construction.

Thank you for your attention!