Square‑root Vélu algorithm for elliptic-curve isogenies#

The square-root Vélu algorithm, also called the √élu algorithm, computes isogenies of elliptic curves in time \(\tilde O(\sqrt\ell)\) rather than naïvely \(O(\ell)\), where \(\ell\) is the degree.

The core idea is to reindex the points in the kernel subgroup in a baby-step-giant-step manner, then use fast resultant computations to evaluate “elliptic polynomials” (see FastEllipticPolynomial) in essentially square-root time.

Based on experiments with Sage version 9.7, the isogeny degree where EllipticCurveHom_velusqrt begins to outperform EllipticCurveIsogeny can be as low as \(\approx 100\), but is typically closer to \(\approx 1000\), depending on the exact situation.

REFERENCES: [BDLS2020]

EXAMPLES:

sage: from sage.schemes.elliptic_curves.hom_velusqrt import EllipticCurveHom_velusqrt
sage: E = EllipticCurve(GF(6666679), [5,5])
sage: K = E(9970, 1003793, 1)
sage: K.order()
10009
sage: phi = EllipticCurveHom_velusqrt(E, K)
sage: phi
Elliptic-curve isogeny (using square-root Vélu) of degree 10009:
  From: Elliptic Curve defined by y^2 = x^3 + 5*x + 5 over Finite Field of size 6666679
  To:   Elliptic Curve defined by y^2 = x^3 + 227975*x + 3596133 over Finite Field of size 6666679
sage: phi.codomain()
Elliptic Curve defined by y^2 = x^3 + 227975*x + 3596133 over Finite Field of size 6666679

Note that the isogeny is usually not identical to the one computed by EllipticCurveIsogeny:

sage: psi = EllipticCurveIsogeny(E, K)
sage: psi
Isogeny of degree 10009
    from Elliptic Curve defined by y^2 = x^3 + 5*x + 5 over Finite Field of size 6666679
    to Elliptic Curve defined by y^2 = x^3 + 5344836*x + 3950273 over Finite Field of size 6666679

However, they are certainly separable isogenies with the same kernel and must therefore be equal up to post-isomorphism:

sage: isos = psi.codomain().isomorphisms(phi.codomain())
sage: sum(iso * psi == phi for iso in isos)
1

Just like EllipticCurveIsogeny, the constructor supports a model keyword argument:

sage: E = EllipticCurve(GF(6666679), [1,1])
sage: K = E(9091, 517864)
sage: phi = EllipticCurveHom_velusqrt(E, K, model='montgomery')
sage: phi
Elliptic-curve isogeny (using square-root Vélu) of degree 2999:
  From: Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 6666679
  To:   Elliptic Curve defined by y^2 = x^3 + 1559358*x^2 + x over Finite Field of size 6666679

Internally, EllipticCurveHom_velusqrt works on short Weierstraß curves, but it performs the conversion automatically:

sage: E = EllipticCurve(GF(101), [1,2,3,4,5])
sage: K = E(1, 2)
sage: K.order()
37
sage: EllipticCurveHom_velusqrt(E, K)
Elliptic-curve isogeny (using square-root Vélu) of degree 37:
  From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Finite Field of size 101
  To:   Elliptic Curve defined by y^2 = x^3 + 66*x + 86 over Finite Field of size 101

However, this does imply not all elliptic curves are supported. Curves without a short Weierstraß model exist in characteristics \(2\) and \(3\):

sage: F.<t> = GF(3^3)
sage: E = EllipticCurve(F, [1,1,1,1,1])
sage: P = E(t^2+2, 1)
sage: P.order()
19
sage: EllipticCurveHom_velusqrt(E, P)
Traceback (most recent call last):
...
NotImplementedError: only implemented for curves having a short Weierstrass model

Furthermore, the implementation is restricted to finite fields, since this appears to be the most relevant application for the square-root Vélu algorithm:

sage: E = EllipticCurve('26b1')
sage: P = E(1,0)
sage: P.order()
7
sage: EllipticCurveHom_velusqrt(E, P)
Traceback (most recent call last):
...
NotImplementedError: only implemented for elliptic curves over finite fields

Note

Some of the methods inherited from EllipticCurveHom compute data whose size is linear in the degree; this includes kernel polynomial and rational maps. In consequence, those methods cannot possibly run in the otherwise advertised square-root complexity, as merely storing the result already takes linear time.

AUTHORS:

  • Lorenz Panny (2022)

class sage.schemes.elliptic_curves.hom_velusqrt.EllipticCurveHom_velusqrt(E, P, *, codomain=None, model=None, Q=None)#

Bases: EllipticCurveHom

This class implements separable odd-degree isogenies of elliptic curves over finite fields using the square-root Vélu algorithm.

The complexity is \(\tilde O(\sqrt{\ell})\) base-field operations, where \(\ell\) is the degree.

REFERENCES: [BDLS2020]

INPUT:

  • E – an elliptic curve over a finite field

  • P – a point on \(E\) of odd order \(\geq 9\)

  • codomain – codomain elliptic curve (optional)

  • model – string (optional); input to compute_model()

  • Q – a point on \(E\) outside \(\langle P\rangle\), or None

EXAMPLES:

sage: from sage.schemes.elliptic_curves.hom_velusqrt import EllipticCurveHom_velusqrt
sage: F.<t> = GF(10009^3)
sage: E = EllipticCurve(F, [t,t])
sage: K = E(2154*t^2 + 5711*t + 2899, 7340*t^2 + 4653*t + 6935)
sage: phi = EllipticCurveHom_velusqrt(E, K); phi
Elliptic-curve isogeny (using square-root Vélu) of degree 601:
  From: Elliptic Curve defined by y^2 = x^3 + t*x + t over Finite Field in t of size 10009^3
  To:   Elliptic Curve defined by y^2 = x^3 + (263*t^2+3173*t+4759)*x + (3898*t^2+6111*t+9443) over Finite Field in t of size 10009^3
sage: phi(K)
(0 : 1 : 0)
sage: P = E(2, 3163*t^2 + 7293*t + 5999)
sage: phi(P)
(6085*t^2 + 855*t + 8720 : 8078*t^2 + 9889*t + 6030 : 1)
sage: Q = E(6, 5575*t^2 + 6607*t + 9991)
sage: phi(Q)
(626*t^2 + 9749*t + 1291 : 5931*t^2 + 8549*t + 3111 : 1)
sage: phi(P + Q)
(983*t^2 + 4894*t + 4072 : 5047*t^2 + 9325*t + 336 : 1)
sage: phi(P) + phi(Q)
(983*t^2 + 4894*t + 4072 : 5047*t^2 + 9325*t + 336 : 1)
dual()#

Return the dual of this square-root Vélu isogeny as an EllipticCurveHom.

Note

The dual is computed by EllipticCurveIsogeny, hence it does not benefit from the square-root Vélu speedup.

EXAMPLES:

sage: E = EllipticCurve(GF(101^2), [1, 1, 1, 1, 1])
sage: K = E.cardinality() // 11 * E.gens()[0]
sage: phi = E.isogeny(K, algorithm='velusqrt'); phi
Elliptic-curve isogeny (using square-root Vélu) of degree 11:
  From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
  To:   Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2
sage: phi.dual()
Isogeny of degree 11 from Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2 to Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
sage: phi.dual() * phi == phi.domain().scalar_multiplication(11)
True
sage: phi * phi.dual() == phi.codomain().scalar_multiplication(11)
True
is_separable()#

Determine whether or not this isogeny is separable.

Since EllipticCurveHom_velusqrt only implements separable isogenies, this method always returns True.

EXAMPLES:

sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
sage: phi = E.isogeny(E((1,2)), algorithm='velusqrt')
sage: phi.is_separable()
True
kernel_polynomial()#

Return the kernel polynomial of this square-root Vélu isogeny.

Note

The data returned by this method has size linear in the degree.

EXAMPLES:

sage: E = EllipticCurve(GF(65537^2,'a'), [5,5])
sage: K = E.cardinality()//31 * E.gens()[0]
sage: phi = E.isogeny(K, algorithm='velusqrt')
sage: h = phi.kernel_polynomial(); h
x^15 + 21562*x^14 + 8571*x^13 + 20029*x^12 + 1775*x^11 + 60402*x^10 + 17481*x^9 + 46543*x^8 + 46519*x^7 + 18590*x^6 + 36554*x^5 + 36499*x^4 + 48857*x^3 + 3066*x^2 + 23264*x + 53937
sage: h == E.isogeny(K).kernel_polynomial()
True
sage: h(K.xy()[0])
0
rational_maps()#

Return the pair of explicit rational maps of this square-root Vélu isogeny as fractions of bivariate polynomials in \(x\) and \(y\).

Note

The data returned by this method has size linear in the degree.

EXAMPLES:

sage: E = EllipticCurve(GF(101^2), [1, 1, 1, 1, 1])
sage: K = (E.cardinality() // 11) * E.gens()[0]
sage: phi = E.isogeny(K, algorithm='velusqrt'); phi
Elliptic-curve isogeny (using square-root Vélu) of degree 11:
  From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
  To:   Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2
sage: phi.rational_maps()
((-17*x^11 - 34*x^10 - 36*x^9 + ... - 29*x^2 - 25*x - 25)/(x^10 + 10*x^9 + 19*x^8 - ... + x^2 + 47*x + 24),
 (-3*x^16 - 6*x^15*y - 48*x^15 + ... - 49*x - 9*y + 46)/(x^15 + 15*x^14 - 35*x^13 - ... + 3*x^2 - 45*x + 47))
scaling_factor()#

Return the Weierstrass scaling factor associated to this square-root Vélu isogeny.

The scaling factor is the constant \(u\) (in the base field) such that \(\varphi^* \omega_2 = u \omega_1\), where \(\varphi: E_1\to E_2\) is this isogeny and \(\omega_i\) are the standard Weierstrass differentials on \(E_i\) defined by \(\mathrm dx/(2y+a_1x+a_3)\).

EXAMPLES:

sage: E = EllipticCurve(GF(101^2), [1, 1, 1, 1, 1])
sage: K = (E.cardinality() // 11) * E.gens()[0]
sage: phi = E.isogeny(K, algorithm='velusqrt', model='montgomery'); phi
Elliptic-curve isogeny (using square-root Vélu) of degree 11:
  From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
  To:   Elliptic Curve defined by y^2 = x^3 + 61*x^2 + x over Finite Field in z2 of size 101^2
sage: phi.scaling_factor()
55
sage: phi.scaling_factor() == phi.formal()[1]
True
x_rational_map()#

Return the \(x\)-coordinate rational map of this square-root Vélu isogeny as a univariate rational function in \(x\).

Note

The data returned by this method has size linear in the degree.

EXAMPLES:

sage: E = EllipticCurve(GF(101^2), [1, 1, 1, 1, 1])
sage: K = (E.cardinality() // 11) * E.gens()[0]
sage: phi = E.isogeny(K, algorithm='velusqrt'); phi
Elliptic-curve isogeny (using square-root Vélu) of degree 11:
  From: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + x + 1 over Finite Field in z2 of size 101^2
  To:   Elliptic Curve defined by y^2 = x^3 + 39*x + 40 over Finite Field in z2 of size 101^2
sage: phi.x_rational_map()
(84*x^11 + 67*x^10 + 65*x^9 + ... + 72*x^2 + 76*x + 76)/(x^10 + 10*x^9 + 19*x^8 + ... + x^2 + 47*x + 24)
sage: phi.x_rational_map() == phi.rational_maps()[0]
True
class sage.schemes.elliptic_curves.hom_velusqrt.FastEllipticPolynomial(E, n, P, Q=None)#

Bases: object

A class to represent and evaluate an elliptic polynomial, and optionally its derivative, in essentially square-root time.

The elliptic polynomials computed by this class are of the form

\[h_S(Z) = \prod_{i\in S} (Z - x(Q + [i]P))\]

where \(P\) is a point of odd order \(n \geq 5\) and \(Q\) is either None, in which case it is assumed to be \(\infty\), or an arbitrary point which is not a multiple of \(P\).

The index set \(S\) is chosen as follows:

  • If \(Q\) is given, then \(S = \{0,1,2,3,...,n-1\}\).

  • If \(Q\) is omitted, then \(S = \{1,3,5,...,n-2\}\). Note that in this case, \(h_{\{1,2,3,...,n-1\}}\) can be computed as \(h_S^2\) since \(n\) is odd.

INPUT:

  • E – an elliptic curve in short Weierstraß form

  • n – an odd integer \(\geq 5\)

  • P – a point on \(E\)

  • Q – a point on \(E\), or None

ALGORITHM: [BDLS2020], Algorithm 2

Note

Currently only implemented for short Weierstraß curves.

EXAMPLES:

sage: from sage.schemes.elliptic_curves.hom_velusqrt import FastEllipticPolynomial
sage: E = EllipticCurve(GF(71), [5,5])
sage: P = E(4, 35)
sage: hP = FastEllipticPolynomial(E, P.order(), P); hP
Fast elliptic polynomial prod(Z - x(i*P) for i in range(1,n,2)) with n = 19, P = (4 : 35 : 1)
sage: hP(7)
19
sage: prod(7 - (i*P).xy()[0] for i in range(1,P.order(),2))
19

Passing \(Q\) changes the index set:

sage: Q = E(0, 17)
sage: hPQ = FastEllipticPolynomial(E, P.order(), P, Q)
sage: hPQ(7)
58
sage: prod(7 - (Q+i*P).xy()[0] for i in range(P.order()))
58

The call syntax has an optional keyword argument derivative, which will make the function return the pair \((h_S(\alpha), h_S'(\alpha))\) instead of just \(h_S(\alpha)\):

sage: hP(7, derivative=True)
(19, 15)
sage: R.<Z> = E.base_field()[]
sage: HP = prod(Z - (i*P).xy()[0] for i in range(1,P.order(),2))
sage: HP
Z^9 + 16*Z^8 + 57*Z^7 + 6*Z^6 + 45*Z^5 + 31*Z^4 + 46*Z^3 + 10*Z^2 + 28*Z + 41
sage: HP(7)
19
sage: HP.derivative()(7)
15
sage: hPQ(7, derivative=True)
(58, 62)
sage: R.<Z> = E.base_field()[]
sage: HPQ = prod(Z - (Q+i*P).xy()[0] for i in range(P.order()))
sage: HPQ
Z^19 + 53*Z^18 + 67*Z^17 + 39*Z^16 + 56*Z^15 + 32*Z^14 + 44*Z^13 + 6*Z^12 + 27*Z^11 + 29*Z^10 + 38*Z^9 + 48*Z^8 + 38*Z^7 + 43*Z^6 + 21*Z^5 + 25*Z^4 + 33*Z^3 + 49*Z^2 + 60*Z
sage: HPQ(7)
58
sage: HPQ.derivative()(7)
62

The input can be an element of any algebra over the base ring:

sage: R.<T> = GF(71)[]
sage: S.<t> = R.quotient(T^2)
sage: hP(7 + t)
15*t + 19