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# look for isogenous curves having j-invariant not in {0, 1728}# Caution: this can take a while!def find_iso(E): for p_test in primes(30): print("trying isogenies of degree %d"%p_test) isos = [] for i in E.isogenies_prime_degree(p_test): print("checking j-invariant of isogeny ", i) jinv = i.codomain().j_invariant() print("j-invariant is ", jinv) if jinv not in (0, 1728): isos.append(i) if len(isos) > 0: for cur_isos in isos: print("found isogeny ", cur_isos) #print("found isogeny ", isos[0]) return isos[0].dual()
return None
def bls12_381_isos(): # BLS12-381 parameters z = -0xd201000000010000 h = (z - 1) ** 2 // 3 q = z ** 4 - z ** 2 + 1 p = z + h * q assert is_prime(p) assert is_prime(q)
# E1 F = GF(p) Ell = EllipticCurve(F, [0, 4]) assert Ell.order() == h * q # E2 F2.<X> = GF(p^2, modulus=[1,0,1]) Ell2 = EllipticCurve(F2, [0, 4 * (1 + X)]) assert Ell2.order() % q == 0
iso_G1 = find_iso(Ell) # an isogeny from E’ to E, Ell_prime = iso_G1.domain() # where this is E’ assert iso_G1(Ell_prime.random_point()).curve() == Ell iso_G2 = find_iso(Ell2) # an isogeny from E2’ to E2, Ell2_prime = iso_G2.domain() # where this is E2’ assert iso_G2(Ell2_prime.random_point()).curve() == Ell2
return (iso_G1, iso_G2)
def bls12_377_isos(): p = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001 # BLS12-377 parameters q = 0x12ab655e9a2ca55660b44d1e5c37b00159aa76fed00000010a11800000000001 z = 0x8508c00000000001 #z = -0xd201000000010000 h = (z - 1) ** 2 // 3 q1 = z ** 4 - z ** 2 + 1 p1 = z + h * q assert(q1 == q) assert(p1 == p) assert is_prime(p) assert is_prime(q)
# E1 F = GF(p) Ell = EllipticCurve(F, [0, 1]) assert Ell.order() == h * q # E2 quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc F.<X> = GF(p)[] F2.<X2> = GF(p^2, modulus=X^2 - quad_non_res) #F2.<X> = GF(p^2, modulus=[1,0,quad_non_res]) B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X2 Ell2 = EllipticCurve(F2, [0, B]) assert Ell2.order() % q == 0
F.<X> = GF(p)[] F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res) B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X6^3 Ell2_6 = EllipticCurve(F6, [0, B]) Ell2_6.order() % q
G1_X = 0x008848defe740a67c8fc6225bf87ff5485951e2caa9d41bb188282c8bd37cb5cd5481512ffcd394eeab9b16eb21be9ef G1_Y = 0x01914a69c5102eff1f674f5d30afeec4bd7fb348ca3e52d96d182ad44fb82305c2fe3d3634a9591afd82de55559c8ea6 #make sure the generator is on the curve X_cx0 = 0x018480be71c785fec89630a2a3841d01c565f071203e50317ea501f557db6b9b71889f52bb53540274e3e48f7c005196 X_cx1 = 0x00ea6040e700403170dc5a51b1b140d5532777ee6651cecbe7223ece0799c9de5cf89984bff76fe6b26bfefa6ea16afe Y_cy0 = 0x00690d665d446f7bd960736bcbb2efb4de03ed7274b49a58e458c282f832d204f2cf88886d8c7c2ef094094409fd4ddf Y_cy1 = 0x00f8169fd28355189e549da3151a70aa61ef11ac3d591bf12463b01acee304c24279b83f5e52270bd9a1cdd185eb8f93
iso_G1 = find_iso(Ell) # an isogeny from E’ to E, Ell_prime = iso_G1.domain() # where this is E’ assert iso_G1(Ell_prime.random_point()).curve() == Ell generate_WBParams_Fq(iso_G1)
# just making sure if there is any isogeny that is defined over F2 up to degree 30 print("searching for isogeny of Ell2 on F2 up to degree 30") iso_G2 = find_iso(Ell2) generate_WBParams_Fq2(iso_G2)
print("searching for isogeny of Ell2 on F6 up to degree 30") iso_G2_F6 = find_iso(Ell2_6) #iso_G2=iso_G2_F6 # an isogeny from E2’ to E2, Ell2_prime = iso_G2.domain() # where this is E2’ assert iso_G2_F6(Ell2_prime.random_point()).curve() == Ell2_6 return (iso_G1, iso_G2)
def trace_endo(P, p2): ParentCurve = P.curve() return P + ParentCurve((P[0]^p2, P[1]^p2)) + ParentCurve((P[0]^(p2^2), P[1]^(p2^2))) def bls12_377_hash_to_G2(e2_p6S_iso, data): p = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001 # BLS12-377 parameters Fp = GF(p) # E2 quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc F.<X> = GF(p)[] F2.<X2> = GF(p^2, modulus=X^2 - quad_non_res) #F2.<X> = GF(p^2, modulus=[1,0,quad_non_res]) B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X2 Ell2 = EllipticCurve(F2, [0, B])
F.<X> = GF(p)[] F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res) B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X6^3 Ell2_6 = EllipticCurve(F6, [0, B])
Fpelm = Fp(hash(data)) Ep2 = e2_p6S_iso.domain() a = Ep2.hyperelliptic_polynomials()[0][1] b = Ep2.hyperelliptic_polynomials()[0][0] X_0 = - (b/a) * ( 1 + 1/(xsi^2*Fpelm^4 + xsi*Fpelm^2))
if Ep2.is_x_coord(X_0): P_p = Ep2.lift_x(X_0) else: P_p = Ep2.lift_x(xsi*Fpelm^2*X_0)
P_F_6 = e2_p6S_iso(P_p)
P = trace_endo(P_F_6, p^2)
x_p = P[0].polynomial().coefficients()[1]*X2 + P[0].polynomial().coefficients()[0] y_p = P[1].polynomial().coefficients()[1]*X2 + P[1].polynomial().coefficients()[0] P_down = Ell2((x_p,y_p))
return P_down # BLS12-377 curve is fully defined by the following set of parameters (coefficient A=0 for all BLS12 curves):
# Base field modulus = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001# B coefficient = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001# Main subgroup order = 0x12ab655e9a2ca55660b44d1e5c37b00159aa76fed00000010a11800000000001# Extension tower:# Fp2 construction:# Fp quadratic non-residue = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc# Fp6/Fp12 construction:# Fp2 cubic non-residue c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000# Fp2 cubic non-residue c1 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001# Twist parameters:# Twist type: D# B coefficient for twist c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000# B coefficient for twist c1 = 0x010222f6db0fd6f343bd03737460c589dc7b4f91cd5fd889129207b63c6bf8000dd39e5c1ccccccd1c9ed9999999999a# Generators:# G1:# X = 0x008848defe740a67c8fc6225bf87ff5485951e2caa9d41bb188282c8bd37cb5cd5481512ffcd394eeab9b16eb21be9ef# Y = 0x01914a69c5102eff1f674f5d30afeec4bd7fb348ca3e52d96d182ad44fb82305c2fe3d3634a9591afd82de55559c8ea6# G2:# X cx0 = 0x018480be71c785fec89630a2a3841d01c565f071203e50317ea501f557db6b9b71889f52bb53540274e3e48f7c005196# X cx1 = 0x00ea6040e700403170dc5a51b1b140d5532777ee6651cecbe7223ece0799c9de5cf89984bff76fe6b26bfefa6ea16afe# Y cy0 = 0x00690d665d446f7bd960736bcbb2efb4de03ed7274b49a58e458c282f832d204f2cf88886d8c7c2ef094094409fd4ddf# Y cy1 = 0x00f8169fd28355189e549da3151a70aa61ef11ac3d591bf12463b01acee304c24279b83f5e52270bd9a1cdd185eb8f93# Pairing parameters:# |x| (miller loop scalar) = 0x8508c00000000001# x is negative = false
# Curve information:
# Base field: q = 258664426012969094010652733694893533536393512754914660539884262666720468348340822774968888139573360124440321458177# Scalar field: r = 8444461749428370424248824938781546531375899335154063827935233455917409239041# valuation(q - 1, 2) = 46# valuation(r - 1, 2) = 47# G1 curve equation: y^2 = x^3 + 1# G2 curve equation: y^2 = x^3 + B, where# B = Fq2(0, 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906)
# e26_order = Ell2_6.order()# for i in primes(30):# if e26_order % i == 0:# print("order is divisable by ", i)
def find_non_square(): quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc F.<X> = GF(p)[] F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res)
xsi = 0 R.<X> = F6[] for i in F6: j = F6.random_element() if not j.is_square(): xsi = j break
return xsi
def find_non_square_low_abs(base_field): for i in base_field: if not i.is_square(): print(i, "is a nonsquare") return i elif not (-i).is_square(): print(-i, "is a nonsquare") return -i
def find_non_square_low_abs_in_F2(extended_field): """
Note that everything in Fp is an square in Fp^2 suppose a in F_p^2 and b in F_p are primitive elements in their respective field. a and b have multiplicative order (p-1)(p+1) and (p-1) suppose b = a^c so b^(p-1) = a^(c)(p-1) = 1 so d(p+1)(p-1) = (c)(p-1) so (p+1)|c => 2|c so then (a^c/2)^2 = b so b has square.
also all degree 2 extensions of Fp are isomorphic. so if you add square root of an element you can find it in the original Fp """
base_field = extended_field.base() for i in base_field: for j in base_field: for k in [(1,1),(-1,1),(1,-1),(-1,-1)]: candidate = extended_field([i*k[0],j*k[1]]) if not candidate.is_square(): print(candidate, "is a nonsquare") return candidate
def generate_WBParams_Fq(Fq_isogeny): print("\nconst PHI_X_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(Fq_isogeny.rational_maps()[0].numerator()))
print("\nconst PHI_X_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(Fq_isogeny.rational_maps()[0].denominator()))
print("\nconst PHI_Y_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(phi2_num(Fq_isogeny)))
print("\nconst PHI_Y_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(Fq_isogeny.rational_maps()[1].denominator()))
def convert_g1_coeff_arrays_to_arkworks(iso_poly): """
convert an array of coefficients in Fp2 to arkwork format """
arkwork_array = '[\n' for cur_deg in range(0, iso_poly.degree() + 1): coeff_selector = [cur_deg] coeff_selector.extend([0]*(len(parent(iso_poly).gens())-1)) arkwork_array += 'field_new!(Fq, \"' + str(iso_poly.coefficient(coeff_selector)) + '\"), \n'
arkwork_array = arkwork_array[: -2] + '\n];' return arkwork_array
def phi2_num(Fq2_isogeny): return (Fq2_isogeny.rational_maps()[1].numerator()/Fq2_isogeny.rational_maps()[1].numerator().variables()[1]).numerator()
def generate_WBParams_Fq2(Fq2_isogeny): print("\nconst PHI_X_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(Fq2_isogeny.rational_maps()[0].numerator()))
print("\nconst PHI_X_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(Fq2_isogeny.rational_maps()[0].denominator()))
print("\nconst PHI_Y_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(phi2_num(Fq2_isogeny)))
print("\nconst PHI_Y_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(Fq2_isogeny.rational_maps()[1].denominator()))
def convert_g2_coeff_arrays_to_arkworks(iso_poly): """
convert an array of coefficients in Fp2 to arkwork format """
arkwork_array = '[\n' for cur_deg in range(0, iso_poly.degree() + 1): elm_poly = iso_poly.coefficient([cur_deg]) assert(elm_poly.is_constant()) elm_poly = elm_poly.constant_coefficient().polynomial() x_gen = parent(elm_poly).gen() arkwork_array += 'field_new!(Fq2, \nfield_new!(Fq, \"' + str(elm_poly.monomial_coefficient(x_gen^0)) + '\"), \nfield_new!(Fq, \"' + str(elm_poly.monomial_coefficient(x_gen^1)) + '\")), \n'
arkwork_array = arkwork_array[: -2] + '\n];' return arkwork_array
#xsi = find_non_square()p = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffcF.<X> = GF(p)[]F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res)xsi = 219316876564715501445845678793720069854490678622108189766720284953606550746049804013989455416277727490975087855145*X6^5 + 189466179801738810887203415755624953151244475240588395433085007196132075694018697411282623497091853518199320679360*X6^4 + 108520825812509855860919714742321990242401369063484109910639742940603686352843166577300516208027451357460988981712*X6^3 + 34112775793303707113282524593132637468394825594735587392051797399858875786055772232469914619100641475357422803395*X6^2 + 24571511079604071320024029432159749780558012896103101565537728634741822079830191849836550641682167566004999041634*X6 + 159529377420408936856843219892213059773340895657856533814435812939320020966386099270143668590340681432634121813716
g1_iso, g2_iso = bls12_377_isos()message = "'I refuse to prove that I exist,' says God, 'for proof denies faith, and without faith I am nothing.'"print(bls12_377_hash_to_G2(g2_iso, message))print(bls12_377_hash_to_G2(g2_iso, message))message = "'if you stick a Babel fish in your ear you can instantly understand anything said to you in any form of language."print(bls12_377_hash_to_G2(g2_iso, message))print(bls12_377_hash_to_G2(g2_iso, message))
phi_x_nom = [F2(257686488674545770403807165703608491821700153538449954828958424244540050698630649531963334687249831352703307010320), 111424637101855455933266154646364284011426845669269128135151076357862608862363255550023430066507374024948118755170*X2 + 42757221749971324771094873984161893488770713619971906650868412147150411626107692517374735631529074208182971223761, 225121846650282460844501132545123914298308844633699320249517374760159461135641190512758348530493533776082794775238*X2 + 182397511129719455005407008314265069548677727690813781407770284951663734172103638427690475141072553074575221965383, 207900273391847649346738694548609379565855077833395139615718913929125517933139374225345876078131155102796477330196*X2 + 57815299880375466472857931022912171296473275308202666945592250229771936329503691945881925004852905385160277065937, 126932856673577834557989832150639712812952235385146245135235966945536973066700604316981524138335070494135486842267*X2 + 137461512605659170682300574841422927080299499340556631216517745531687218204749421916211266297116812992348911035943, 239220883081126775212690475926873251881459118214247205003758844846022330659109168499739536107988232402392010148915*X2 + 46999088780962505530452862218145506822520603893157196954987630663398814970386085186714696215299498862118704356116, 134745690770497208213126494241047670387239583870118646766457825646403640347226342104549240830903829682059982326727*X2 + 139197805910452438551419010260519164074855817417063470117576949912038778696248952069759607969690314639425179841530, 13030331252003455924520111292282390143014750833628314421882396445822808054003584720817672388894671968339344750750*X2 + 205538184807792915395400814312734290381492118179294333381188081024280220461983210331268164749465891791515156713363, 140302880976816076816721836344737338071783799730513599397010923016385668972882234545823525686327074617973208860421*X2 + 142477190388553987083175296028785369398580520754966744245694134692905987471896623788336582063245356609279657681007, 139097554917981907719834170888130444861465074977932740823078016609751284491153969166862091045221603146530147099779*X2 + 224387508661509885922784938707800296916307265709615639996851552571542396031861010679067114387017955365668256976459, 168051662766288660516992486993594951443897767271245608184500932350745292043277998040372769419920145370014740823389*X2 + 205121513164720886728676669276499362266139599046368626660027529707299170205603076104821498121463983514654737906998, 75932324143801627944771670190157701465835368459121478812142087933983158148741884038298732552979549182443318608554*X2 + 121805396188033590038927712795579559087103093135515082191363074395734682090746320669474388875423586929711608364404, 71494892585734577638768956295356005795556178117525317604450863442514001295461407965931789120761632296953990284809*X2 + 129405180560592211762572081137246817341681076084973348177153469170177032429929182033872985192027066614650754524309, 7531651998638745138624528632013700806848273210661661935097722938692087106885113623803921367479582718152259987647*X2 + 33702103741469458207888758659069786556456463428431771947788685622778743201883736580054112022350187936628758294279, 233542978062801907184831603532041452683131033894029390005010898310417350737942550513935694209129350870817277172583*X2 + 188864327416940344326229259749844123042825201085435432627935318423081174922012610651352346391417830722940140585036, 55517007457989983858891530398020104232682844055600438506715652841000901588962918721851719951820185832313468320365*X2 + 18662780664429933771192510151421557554668611953190652639195940454142648933454488952361465779870877163142807899993, 97587419381711441517698658129839468394457401157021696651597713140291602428957555768352336399994179660880524139808*X2 + 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