1# look for isogenous curves having j-invariant not in {0, 1728}
  2# Caution: this can take a while!
  3def find_iso(E):
  4    for p_test in primes(30):
  5        print("trying isogenies of degree %d"%p_test)
  6        isos = []
  7        for i in E.isogenies_prime_degree(p_test):            
  8            print("checking j-invariant of isogeny ", i)
  9            jinv = i.codomain().j_invariant()
 10            print("j-invariant is ", jinv)
 11            if jinv not in (0, 1728):
 12                isos.append(i)
 13        
 14        if len(isos) > 0:
 15            for cur_isos in isos:
 16                print("found isogeny ", cur_isos)
 17            #print("found isogeny ", isos[0])
 18            return isos[0].dual()
 19
 20    return None
 21
 22def bls12_381_isos():
 23    # BLS12-381 parameters
 24    z = -0xd201000000010000
 25    h = (z - 1) ** 2 // 3
 26    q = z ** 4 - z ** 2 + 1
 27    p = z + h * q
 28    assert is_prime(p)
 29    assert is_prime(q)
 30
 31    # E1
 32    F = GF(p)
 33    Ell = EllipticCurve(F, [0, 4])
 34    assert Ell.order() == h * q
 35    # E2
 36    F2.<X> = GF(p^2, modulus=[1,0,1])
 37    Ell2 = EllipticCurve(F2, [0, 4 * (1 + X)])
 38    assert Ell2.order() % q == 0
 39
 40    iso_G1 = find_iso(Ell)
 41    # an isogeny from E’ to E,
 42    Ell_prime = iso_G1.domain()
 43    # where this is E’
 44    assert iso_G1(Ell_prime.random_point()).curve() == Ell
 45    iso_G2 = find_iso(Ell2)
 46    # an isogeny from E2’ to E2,
 47    Ell2_prime = iso_G2.domain()
 48    # where this is E2’
 49    assert iso_G2(Ell2_prime.random_point()).curve() == Ell2
 50
 51    return (iso_G1, iso_G2)
 52
 53def bls12_377_isos():
 54    p = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001    # BLS12-377 parameters
 55    q = 0x12ab655e9a2ca55660b44d1e5c37b00159aa76fed00000010a11800000000001
 56    z = 0x8508c00000000001
 57    #z = -0xd201000000010000
 58    h = (z - 1) ** 2 // 3
 59    q1 = z ** 4 - z ** 2 + 1
 60    p1 = z + h * q
 61    assert(q1 == q)
 62    assert(p1 == p)
 63    assert is_prime(p)
 64    assert is_prime(q)
 65
 66    # E1
 67    F = GF(p)
 68    Ell = EllipticCurve(F, [0, 1])
 69    assert Ell.order() == h * q
 70    # E2
 71    quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc
 72    F.<X> = GF(p)[]
 73    F2.<X2> = GF(p^2, modulus=X^2 - quad_non_res)
 74    #F2.<X> = GF(p^2, modulus=[1,0,quad_non_res])
 75    B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X2
 76    Ell2 = EllipticCurve(F2, [0, B])
 77    assert Ell2.order() % q == 0
 78
 79    F.<X> = GF(p)[]
 80    F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res)
 81    B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X6^3
 82    Ell2_6 = EllipticCurve(F6, [0, B])
 83    Ell2_6.order() % q
 84
 85    G1_X = 0x008848defe740a67c8fc6225bf87ff5485951e2caa9d41bb188282c8bd37cb5cd5481512ffcd394eeab9b16eb21be9ef
 86    G1_Y = 0x01914a69c5102eff1f674f5d30afeec4bd7fb348ca3e52d96d182ad44fb82305c2fe3d3634a9591afd82de55559c8ea6
 87    #make sure the generator is on the curve
 88    X_cx0 = 0x018480be71c785fec89630a2a3841d01c565f071203e50317ea501f557db6b9b71889f52bb53540274e3e48f7c005196
 89    X_cx1 = 0x00ea6040e700403170dc5a51b1b140d5532777ee6651cecbe7223ece0799c9de5cf89984bff76fe6b26bfefa6ea16afe
 90    Y_cy0 = 0x00690d665d446f7bd960736bcbb2efb4de03ed7274b49a58e458c282f832d204f2cf88886d8c7c2ef094094409fd4ddf
 91    Y_cy1 = 0x00f8169fd28355189e549da3151a70aa61ef11ac3d591bf12463b01acee304c24279b83f5e52270bd9a1cdd185eb8f93
 92
 93    iso_G1 = find_iso(Ell)
 94    # an isogeny from E’ to E,
 95    Ell_prime = iso_G1.domain()
 96    # where this is E’
 97    assert iso_G1(Ell_prime.random_point()).curve() == Ell
 98    generate_WBParams_Fq(iso_G1)
 99
100    # just making sure if there is any isogeny that is defined over F2 up to degree 30
101    print("searching for isogeny of Ell2 on F2 up to degree 30")
102    iso_G2 = find_iso(Ell2)
103    generate_WBParams_Fq2(iso_G2)
104
105    print("searching for isogeny of Ell2 on F6 up to degree 30")
106    iso_G2_F6 = find_iso(Ell2_6)
107    #iso_G2=iso_G2_F6
108    # an isogeny from E2’ to E2,
109    Ell2_prime = iso_G2.domain()
110    # where this is E2’
111    assert iso_G2_F6(Ell2_prime.random_point()).curve() == Ell2_6    
112    return (iso_G1, iso_G2)
113
114def trace_endo(P, p2):
115    ParentCurve = P.curve()
116    return P + ParentCurve((P[0]^p2, P[1]^p2)) + ParentCurve((P[0]^(p2^2), P[1]^(p2^2)))
117    
118def bls12_377_hash_to_G2(e2_p6S_iso, data):
119    p = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001    # BLS12-377 parameters
120    Fp = GF(p)
121    # E2
122    quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc
123    F.<X> = GF(p)[]
124    F2.<X2> = GF(p^2, modulus=X^2 - quad_non_res)
125    #F2.<X> = GF(p^2, modulus=[1,0,quad_non_res])
126    B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X2
127    Ell2 = EllipticCurve(F2, [0, B])
128
129    F.<X> = GF(p)[]
130    F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res)
131    B = 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906 * X6^3
132    Ell2_6 = EllipticCurve(F6, [0, B])
133
134    Fpelm = Fp(hash(data))
135            
136    Ep2 = e2_p6S_iso.domain()
137    a = Ep2.hyperelliptic_polynomials()[0][1]
138    b = Ep2.hyperelliptic_polynomials()[0][0]
139    
140    X_0 = - (b/a) * ( 1 + 1/(xsi^2*Fpelm^4 + xsi*Fpelm^2))
141
142    if Ep2.is_x_coord(X_0):
143        P_p = Ep2.lift_x(X_0)
144    else:
145        P_p = Ep2.lift_x(xsi*Fpelm^2*X_0)
146
147    P_F_6 = e2_p6S_iso(P_p)
148
149    P = trace_endo(P_F_6, p^2)
150
151    x_p = P[0].polynomial().coefficients()[1]*X2 + P[0].polynomial().coefficients()[0]
152    y_p = P[1].polynomial().coefficients()[1]*X2 + P[1].polynomial().coefficients()[0]
153    
154    P_down = Ell2((x_p,y_p))
155
156    return P_down
157    
158# BLS12-377 curve is fully defined by the following set of parameters (coefficient A=0 for all BLS12 curves):
159
160# Base field modulus = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001
161# B coefficient = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
162# Main subgroup order = 0x12ab655e9a2ca55660b44d1e5c37b00159aa76fed00000010a11800000000001
163# Extension tower:
164# Fp2 construction:
165# Fp quadratic non-residue = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc
166# Fp6/Fp12 construction:
167# Fp2 cubic non-residue c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
168# Fp2 cubic non-residue c1 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001
169# Twist parameters:
170# Twist type: D
171# B coefficient for twist c0 = 0x000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
172# B coefficient for twist c1 = 0x010222f6db0fd6f343bd03737460c589dc7b4f91cd5fd889129207b63c6bf8000dd39e5c1ccccccd1c9ed9999999999a
173# Generators:
174# G1:
175# X = 0x008848defe740a67c8fc6225bf87ff5485951e2caa9d41bb188282c8bd37cb5cd5481512ffcd394eeab9b16eb21be9ef
176# Y = 0x01914a69c5102eff1f674f5d30afeec4bd7fb348ca3e52d96d182ad44fb82305c2fe3d3634a9591afd82de55559c8ea6
177# G2:
178# X cx0 = 0x018480be71c785fec89630a2a3841d01c565f071203e50317ea501f557db6b9b71889f52bb53540274e3e48f7c005196
179# X cx1 = 0x00ea6040e700403170dc5a51b1b140d5532777ee6651cecbe7223ece0799c9de5cf89984bff76fe6b26bfefa6ea16afe
180# Y cy0 = 0x00690d665d446f7bd960736bcbb2efb4de03ed7274b49a58e458c282f832d204f2cf88886d8c7c2ef094094409fd4ddf
181# Y cy1 = 0x00f8169fd28355189e549da3151a70aa61ef11ac3d591bf12463b01acee304c24279b83f5e52270bd9a1cdd185eb8f93
182# Pairing parameters:
183# |x| (miller loop scalar) = 0x8508c00000000001
184# x is negative = false
185
186# Curve information:
187
188# Base field: q = 258664426012969094010652733694893533536393512754914660539884262666720468348340822774968888139573360124440321458177
189# Scalar field: r = 8444461749428370424248824938781546531375899335154063827935233455917409239041
190# valuation(q - 1, 2) = 46
191# valuation(r - 1, 2) = 47
192# G1 curve equation: y^2 = x^3 + 1
193# G2 curve equation: y^2 = x^3 + B, where
194# B = Fq2(0, 155198655607781456406391640216936120121836107652948796323930557600032281009004493664981332883744016074664192874906)
195
196
197# e26_order = Ell2_6.order()
198# for i in primes(30):
199#     if e26_order % i == 0:
200#         print("order is divisable by ", i)
201
202def find_non_square():
203    quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc
204    F.<X> = GF(p)[]
205    F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res)
206
207    xsi = 0
208    R.<X> = F6[]
209    for i in F6:
210        j = F6.random_element()
211        if not j.is_square():
212            xsi = j
213            break
214
215    return xsi
216
217def find_non_square_low_abs(base_field):
218    for i in base_field:
219        if not i.is_square():
220           print(i, "is a nonsquare")
221           return i
222        elif not (-i).is_square():
223           print(-i, "is a nonsquare")
224           return -i
225
226def find_non_square_low_abs_in_F2(extended_field):
227    """
228    Note that everything in Fp is an square in Fp^2
229    suppose a in F_p^2 and b in F_p are primitive elements 
230    in their respective field. 
231    a and b have multiplicative order (p-1)(p+1) and (p-1) 
232    suppose b = a^c so b^(p-1) = a^(c)(p-1) = 1 so
233    d(p+1)(p-1) = (c)(p-1) so (p+1)|c => 2|c so then
234    (a^c/2)^2 = b so b has square.
235
236    also all degree 2 extensions of Fp are isomorphic. so 
237    if you add square root of an element you can find it 
238    in the original Fp
239    """
240    base_field = extended_field.base()
241    for i in base_field:
242        for j in base_field:
243            for k in [(1,1),(-1,1),(1,-1),(-1,-1)]:
244                candidate = extended_field([i*k[0],j*k[1]])
245                if not candidate.is_square():
246                    print(candidate, "is a nonsquare")
247                    return candidate
248
249def generate_WBParams_Fq(Fq_isogeny):
250    print("\nconst PHI_X_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(Fq_isogeny.rational_maps()[0].numerator()))
251
252    print("\nconst PHI_X_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(Fq_isogeny.rational_maps()[0].denominator()))
253
254    print("\nconst PHI_Y_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(phi2_num(Fq_isogeny)))
255
256    print("\nconst PHI_Y_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g1_coeff_arrays_to_arkworks(Fq_isogeny.rational_maps()[1].denominator()))
257
258    
259def convert_g1_coeff_arrays_to_arkworks(iso_poly):
260    """
261    convert an array of coefficients in Fp2 to arkwork format
262    """
263    arkwork_array = '[\n'
264    for cur_deg in range(0, iso_poly.degree() + 1):
265        coeff_selector = [cur_deg]
266        coeff_selector.extend([0]*(len(parent(iso_poly).gens())-1))
267        arkwork_array += 'field_new!(Fq, \"' + str(iso_poly.coefficient(coeff_selector)) + '\"), \n'
268
269    arkwork_array = arkwork_array[: -2] + '\n];'
270    return arkwork_array
271
272def phi2_num(Fq2_isogeny):
273    return (Fq2_isogeny.rational_maps()[1].numerator()/Fq2_isogeny.rational_maps()[1].numerator().variables()[1]).numerator()
274
275
276def generate_WBParams_Fq2(Fq2_isogeny):
277    print("\nconst PHI_X_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(Fq2_isogeny.rational_maps()[0].numerator()))
278
279    print("\nconst PHI_X_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(Fq2_isogeny.rational_maps()[0].denominator()))
280
281    print("\nconst PHI_Y_NOM: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(phi2_num(Fq2_isogeny)))
282
283    print("\nconst PHI_Y_DEN: &'static [<Self::IsogenousCurve as ModelParameters>::BaseField] = &"+convert_g2_coeff_arrays_to_arkworks(Fq2_isogeny.rational_maps()[1].denominator()))
284
285def convert_g2_coeff_arrays_to_arkworks(iso_poly):
286    """
287    convert an array of coefficients in Fp2 to arkwork format
288    """
289    arkwork_array = '[\n'
290    for cur_deg in range(0, iso_poly.degree() + 1):
291        elm_poly = iso_poly.coefficient([cur_deg])
292        assert(elm_poly.is_constant())
293        elm_poly = elm_poly.constant_coefficient().polynomial()
294        x_gen = parent(elm_poly).gen()        
295        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'
296
297    arkwork_array = arkwork_array[: -2] + '\n];'
298    return arkwork_array
299
300#xsi = find_non_square()
301p = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508c00000000001
302quad_non_res = 0x01ae3a4617c510eac63b05c06ca1493b1a22d9f300f5138f1ef3622fba094800170b5d44300000008508bffffffffffc
303F.<X> = GF(p)[]
304F6.<X6> = GF(p^6, modulus=X^6 - quad_non_res)
305xsi = 219316876564715501445845678793720069854490678622108189766720284953606550746049804013989455416277727490975087855145*X6^5 + 189466179801738810887203415755624953151244475240588395433085007196132075694018697411282623497091853518199320679360*X6^4 + 108520825812509855860919714742321990242401369063484109910639742940603686352843166577300516208027451357460988981712*X6^3 + 34112775793303707113282524593132637468394825594735587392051797399858875786055772232469914619100641475357422803395*X6^2 + 24571511079604071320024029432159749780558012896103101565537728634741822079830191849836550641682167566004999041634*X6 + 159529377420408936856843219892213059773340895657856533814435812939320020966386099270143668590340681432634121813716
306
307g1_iso, g2_iso = bls12_377_isos()
308message = "'I refuse to prove that I exist,' says God, 'for proof denies faith, and without faith I am nothing.'"
309print(bls12_377_hash_to_G2(g2_iso, message))
310print(bls12_377_hash_to_G2(g2_iso, message))
311message = "'if you stick a Babel fish in your ear you can instantly understand anything said to you in any form of language."
312print(bls12_377_hash_to_G2(g2_iso, message))
313print(bls12_377_hash_to_G2(g2_iso, message))
314
315
316
317
318
319
320
321phi_x_nom = [F2(257686488674545770403807165703608491821700153538449954828958424244540050698630649531963334687249831352703307010320), 111424637101855455933266154646364284011426845669269128135151076357862608862363255550023430066507374024948118755170*X2 + 42757221749971324771094873984161893488770713619971906650868412147150411626107692517374735631529074208182971223761, 225121846650282460844501132545123914298308844633699320249517374760159461135641190512758348530493533776082794775238*X2 + 182397511129719455005407008314265069548677727690813781407770284951663734172103638427690475141072553074575221965383, 207900273391847649346738694548609379565855077833395139615718913929125517933139374225345876078131155102796477330196*X2 + 57815299880375466472857931022912171296473275308202666945592250229771936329503691945881925004852905385160277065937, 126932856673577834557989832150639712812952235385146245135235966945536973066700604316981524138335070494135486842267*X2 + 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142477190388553987083175296028785369398580520754966744245694134692905987471896623788336582063245356609279657681007, 139097554917981907719834170888130444861465074977932740823078016609751284491153969166862091045221603146530147099779*X2 + 224387508661509885922784938707800296916307265709615639996851552571542396031861010679067114387017955365668256976459, 168051662766288660516992486993594951443897767271245608184500932350745292043277998040372769419920145370014740823389*X2 + 205121513164720886728676669276499362266139599046368626660027529707299170205603076104821498121463983514654737906998, 75932324143801627944771670190157701465835368459121478812142087933983158148741884038298732552979549182443318608554*X2 + 121805396188033590038927712795579559087103093135515082191363074395734682090746320669474388875423586929711608364404, 71494892585734577638768956295356005795556178117525317604450863442514001295461407965931789120761632296953990284809*X2 + 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