selecting class polynomials for the generation of elliptic curves
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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.
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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
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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
pcp
log62.044.0 pc
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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
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Class field polynomials
Alternative class field polynomials:1) Weber polynomials2) MD,l(x) polynomials3) MD,p1,p2(x) polynomials or Double eta polynomials4) 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/4if D 0 mod 4
d = Dif 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
pD
pD
D 11 mod 24
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Hilbert polynomials
))(()( jxxH D
aDb
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.
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Weber polynomials
l
D lgxxW ))(()(
aDbl
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
))(()(,
ADB
Q 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,
ADB
Q 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
pD
12
pD
)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
aDb
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
ihia2,...,0
logmax
Bit precision for the Hilbert polynomials:
],,[
12ln
33)(PrCBA 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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mfr
jfjf
PHPH
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.
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],,[
12ln)( CBA AD
frm
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.
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0)276(),( 183612 HTTTHTT RRRRRR
Precision Estimates
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Precision Estimates
23
Precision Estimates
24
Experiments
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Construction of polynomials (bit prec.)
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Construction of polynomials (bit prec.)
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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!
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