#!/usr/bin/sage # -*- mode: python ; -*- from sage.all import * import hashlib import itertools from hashlib import sha256 p = 52435875175126190479447740508185965837690552500527637822603658699938581184513 F = FiniteField(p) # anemoi is from their repo COST_ALPHA = { 3 : 2, 5 : 3, 7 : 4, 9 : 4, 11 : 5, 13 : 5, 15 : 5, 17 : 5, 19 : 6, 21 : 6, 23 : 6, 25 : 6, 27 : 6, 29 : 7, 31 : 7, 33 : 6, 35 : 7, 37 : 7, 39 : 7, 41 : 7, 43 : 7, 45 : 7, 47 : 8, 49 : 7, 51 : 7, 53 : 8, 55 : 8, 57 : 8, 59 : 8, 61 : 8, 63 : 8, 65 : 7, 67 : 8, 69 : 8, 71 : 9, 73 : 8, 75 : 8, 77 : 8, 79 : 9, 81 : 8, 83 : 8, 85 : 8, 87 : 9, 89 : 9, 91 : 9, 93 : 9, 95 : 9, 97 : 8, 99 : 8, 101 : 9, 103 : 9, 105 : 9, 107 : 9, 109 : 9, 111 : 9, 113 : 9, 115 : 9, 117 : 9, 119 : 9, 121 : 9, 123 : 9, 125 : 9, 127 : 10, } ALPHA_BY_COST = { c : [x for x in range(3, 128, 2) if COST_ALPHA[x] == c] for c in range(2, 11) } PI_0 = 1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679 PI_1 = 8214808651328230664709384460955058223172535940812848111745028410270193852110555964462294895493038196 def get_prime(N): result = (1 << N) - 1 while not is_prime(result): result -= 2 return result def get_n_rounds(s, l, alpha): r = 0 complexity = 0 kappa = {3:1, 5:2, 7:4, 9:7, 11:9} assert alpha in kappa while complexity < 2**s: r += 1 complexity = binomial( 4*l*r + kappa[alpha], 2*l*r )**2 r += 2 # considering the second model r += min(5,l+1) # security margin return max(8, r) # Linear layer generation def is_mds(m): # Uses the Laplace expansion of the determinant to calculate the (m+1)x(m+1) minors in terms of the mxm minors. # Taken from https://github.com/mir-protocol/hash-constants/blob/master/mds_search.sage. # 1-minors are just the elements themselves if any(any(r == 0 for r in row) for row in m): return False N = m.nrows() assert m.is_square() and N >= 2 det_cache = m # Calculate all the nxn minors of m: for n in range(2, N+1): new_det_cache = dict() for rows in itertools.combinations(range(N), n): for cols in itertools.combinations(range(N), n): i, *rs = rows # Laplace expansion along row i det = 0 for j in range(n): # pick out c = column j; the remaining columns are in cs c = cols[j] cs = cols[:j] + cols[j+1:] # Look up the determinant from the previous iteration # and multiply by -1 if j is odd cofactor = det_cache[(*rs, *cs)] if j % 2 == 1: cofactor = -cofactor # update the determinant with the j-th term det += m[i, c] * cofactor if det == 0: return False new_det_cache[(*rows, *cols)] = det det_cache = new_det_cache return True def M_2(x_input, b): x = x_input[:] x[0] += b*x[1] x[1] += b*x[0] return x def M_3(x_input, b): x = x_input[:] t = x[0] + b*x[2] x[2] += x[1] x[2] += b*x[0] x[0] = t + x[2] x[1] += t return x def M_4(x_input, b): x = x_input[:] x[0] += x[1] x[2] += x[3] x[3] += b*x[0] x[1] = b*(x[1] + x[2]) x[0] += x[1] x[2] += b*x[3] x[1] += x[2] x[3] += x[0] return x def lfsr(x_input, b): x = x_input[:] l = len(x) for r in range(0, l): t = sum(b**(2**i) * x[i] for i in range(0, l)) x = x[1:] + [t] return x def circulant_mds_matrix(field, l, coeff_upper_limit=None): if coeff_upper_limit == None: coeff_upper_limit = l+1 assert(coeff_upper_limit > l) for v in itertools.combinations_with_replacement(range(1,coeff_upper_limit), l): mat = matrix.circulant(list(v)).change_ring(field) if is_mds(mat): return(mat) # In some cases, the method won't return any valid matrix, # hence the need to increase the limit further. return circulant_mds_matrix(field, l, coeff_upper_limit+1) def get_mds(field, l): if l == 1: return identity_matrix(field, 1) if l <= 4: # low addition case a = field.multiplicative_generator() b = field.one() t = 0 while True: # we construct the matrix mat = [] b = b*a t += 1 for i in range(0, l): x_i = [field.one() * (j == i) for j in range(0, l)] if l == 2: mat.append(M_2(x_i, b)) elif l == 3: mat.append(M_3(x_i, b)) elif l == 4: mat.append(M_4(x_i, b)) mat = Matrix(field, l, l, mat).transpose() if is_mds(mat): return mat else: # circulant matrix case return circulant_mds_matrix(field, l) # AnemoiPermutation class class AnemoiPermutation: def __init__(self, q=None, alpha=None, mat=None, n_rounds=None, n_cols=1, security_level=128): if q == None: raise Exception("The characteristic of the field must be specified!") self.q = q self.prime_field = is_prime(q) # if true then we work over a # prime field with # characteristic just under # 2**N, otherwise the # characteristic is 2**self self.n_cols = n_cols # the number of parallel S-boxes in each round self.security_level = security_level # initializing the other variables in the state: # - q is the characteristic of the field # - g is a generator of the multiplicative subgroup # - alpha is the main exponent (in the center of the Flystel) # - beta is the coefficient in the quadratic subfunction # - gamma is the constant in the second quadratic subfunction # - QUAD is the secondary (quadratic) exponent # - from_field is a function mapping field elements to integers # - to_field is a function mapping integers to field elements self.F = GF(self.q) if self.prime_field: if alpha != None: if gcd(alpha, self.q-1) != 1: raise Exception("alpha should be co-prime with the characteristic!") else: self.alpha = alpha else: self.alpha = 3 while gcd(self.alpha, self.q-1) != 1: self.alpha += 1 self.QUAD = 2 self.to_field = lambda x : self.F(x) self.from_field = lambda x : Integer(x) else: self.alpha = 3 self.QUAD = 3 self.to_field = lambda x : self.F.fetch_int(x) self.from_field = lambda x : x.integer_representation() self.g = self.F.multiplicative_generator() self.beta = self.g self.delta = self.g**(-1) self.alpha_inv = inverse_mod(self.alpha, self.q-1) # total number of rounds if n_rounds != None: self.n_rounds = n_rounds else: self.n_rounds = get_n_rounds(self.security_level, self.n_cols, self.alpha) # Choosing constants: self.C and self.D are built from the # digits of pi using an open butterfly self.C = [] self.D = [] pi_F_0 = self.to_field(PI_0 % self.q) pi_F_1 = self.to_field(PI_1 % self.q) for r in range(0, self.n_rounds): pi_0_r = pi_F_0**r self.C.append([]) self.D.append([]) for i in range(0, self.n_cols): pi_1_i = pi_F_1**i pow_alpha = (pi_0_r + pi_1_i)**self.alpha self.C[r].append(self.g * (pi_0_r)**2 + pow_alpha) self.D[r].append(self.g * (pi_1_i)**2 + pow_alpha + self.delta) self.mat = get_mds(self.F, self.n_cols) def __str__(self): result = "Anemoi instance over F_{:d} ({}), n_rounds={:d}, n_cols={:d}, s={:d}".format( self.q, "odd prime field" if self.prime_field else "characteristic 2", self.n_rounds, self.n_cols, self.security_level ) result += "\nalpha={}, beta={}, \ndelta={}\nM_x=\n{}\ninv_alpha={}\n".format( self.alpha, self.beta, self.delta, self.mat, self.alpha_inv ) result += "C={}\nD={}".format( [[self.from_field(x) for x in self.C[r]] for r in range(0, self.n_rounds)], [[self.from_field(x) for x in self.D[r]] for r in range(0, self.n_rounds)], ) return result # !SECTION! Sub-components def evaluate_sbox(self, _x, _y): x, y = _x, _y x -= self.beta*y**self.QUAD y -= x**self.alpha_inv x += self.beta*y**self.QUAD + self.delta return x, y def linear_layer(self, _x, _y): x, y = _x[:], _y[:] x = self.mat*vector(x) y = self.mat*vector(y[1:] + [y[0]]) # Pseudo-Hadamard transform on each (x,y) pair y += x x += y return list(x), list(y) # !SECTION! Evaluation def eval_with_intermediate_values(self, _x, _y): x, y = _x[:], _y[:] result = [[x[:], y[:]]] for r in range(0, self.n_rounds): for i in range(0, self.n_cols): x[i] += self.C[r][i] y[i] += self.D[r][i] x, y = self.linear_layer(x, y) for i in range(0, self.n_cols): x[i], y[i] = self.evaluate_sbox(x[i], y[i]) result.append([x[:], y[:]]) # final call to the linear layer x, y = self.linear_layer(x, y) result.append([x[:], y[:]]) return result def input_size(self): return 2*self.n_cols def __call__(self, _x): if len(_x) != self.input_size(): raise Exception("wrong input size!") else: x, y = _x[:self.n_cols], _x[self.n_cols:] u, v = self.eval_with_intermediate_values(x, y)[-1] return u + v # concatenation, not a sum # !SECTION! Writing full system of equations def get_polynomial_variables(self): x_vars = [] y_vars = [] all_vars = [] for r in range(0, self.n_rounds+1): x_vars.append(["X{:02d}{:02d}".format(r, i) for i in range(0, self.n_cols)]) y_vars.append(["Y{:02d}{:02d}".format(r, i) for i in range(0, self.n_cols)]) all_vars += x_vars[-1] all_vars += y_vars[-1] pol_ring = PolynomialRing(self.F, (self.n_rounds+1)*2*self.n_cols, all_vars) pol_gens = pol_ring.gens() result = {"X" : [], "Y" : []} for r in range(0, self.n_rounds+1): result["X"].append([]) result["Y"].append([]) for i in range(0, self.n_cols): result["X"][r].append(pol_gens[self.n_cols*2*r + i]) result["Y"][r].append(pol_gens[self.n_cols*2*r + i + self.n_cols]) return result def verification_polynomials(self, pol_vars): equations = [] for r in range(0, self.n_rounds): # the outputs of the open flystel are the state variables x, y at round r+1 u = pol_vars["X"][r+1] v = pol_vars["Y"][r+1] # the inputs of the open flystel are the state variables # x, y at round r after undergoing the constant addition # and the linear layer x, y = pol_vars["X"][r], pol_vars["Y"][r] x = [x[i] + self.C[r][i] for i in range(0, self.n_cols)] y = [y[i] + self.D[r][i] for i in range(0, self.n_cols)] x, y = self.linear_layer(x, y) for i in range(0, self.n_cols): equations.append( (y[i]-v[i])**self.alpha + self.beta*y[i]**self.QUAD - x[i] ) equations.append( (y[i]-v[i])**self.alpha + self.beta*v[i]**self.QUAD + self.delta - u[i] ) return equations def print_verification_polynomials(self): p_vars = self.get_polynomial_variables() eqs = self.verification_polynomials(p_vars) variables_string = "" for r in range(0, self.n_rounds+1): variables_string += str(p_vars["X"][r])[1:-1] + "," + str(p_vars["Y"][r])[1:-1] + "," print(variables_string[:-1].replace(" ", "")) print(self.q) for f in eqs: print(f) # !SECTION! Modes of operation def jive(P, b, _x): if b < 2: raise Exception("b must be at least equal to 2") if P.input_size() % b != 0: raise Exception("b must divide the input size!") c = P.input_size()/b # length of the compressed output # Output size check: we allow the output size to be 3 bits shorter than # the theoretical target, as commonly used finite fields usually have a # characteristic size slightly under 2**256. if c * P.F.cardinality().nbits() < 2 * P.security_level - 3: raise Exception(f"digest size is too small for the targeted security level!") x = _x[:] u = P(x) compressed = [] for i in range(0, int(c)): compressed.append(sum(x[int(i+c*j)] + u[int(i+c*j)] for j in range(0, int(b)))) return compressed A_2 = AnemoiPermutation(q=p, alpha=5, n_rounds=None, n_cols=1, security_level=128) A_4 = AnemoiPermutation(q=p, alpha=5, n_rounds=None, n_cols=2, security_level=128) A_16 = AnemoiPermutation(q=p, alpha=5, n_rounds=None, n_cols=8, security_level=128) def anemoi(state): if len(state) == 2: return jive(A_2,2,state)[0] if len(state) == 4: return jive(A_4,4,state)[0] if len(state) == 16: return jive(A_16,16,state)[0] def poseidon(state): if len(state) == 2: original_state = state cst = poseidon_round_constant_2_to_1() state = poseidon_linear_layer_2_to_1(state) for i in range(4): for j in range(2): state[j] += cst[2*i+j] state[j] = state[j]**5 state = poseidon_linear_layer_2_to_1(state) for i in range(56): state[0] += cst[i + 8] state[0] = state[0]**5 state = poseidon_linear_layer_2_to_1(state) for i in range(4): for j in range(2): state[j] += cst[64 + i*2 + j] state[j] = state[j]**5 state = poseidon_linear_layer_2_to_1(state) return state[0] + state[1] + original_state[0] + original_state[1] if len(state) == 4: original_state = state cst = poseidon_round_constant_4_to_1() state = poseidon_external_linear_layer_4_to_1(state) for i in range(4): for j in range(4): state[j] += cst[4*i+j] state[j] = state[j]**5 state = poseidon_external_linear_layer_4_to_1(state) for i in range(56): state[0] += cst[i + 16] state[0] = state[0]**5 state = poseidon_internal_linear_layer_4_to_1(state) for i in range(4): for j in range(4): state[j] += cst[72 + i*4 + j] state[j] = state[j]**5 state = poseidon_external_linear_layer_4_to_1(state) h = F(0) for i in range(4): h += state[i] + original_state[i] return h if len(state) == 16: original_state = state cst = poseidon_round_constant_16_to_1() state = poseidon_external_linear_layer_16_to_1(state) for i in range(4): for j in range(16): state[j] += cst[16*i+j] state[j] = state[j]**5 state = poseidon_external_linear_layer_16_to_1(state) for i in range(57): state[0] += cst[i + 64] state[0] = state[0]**5 state = poseidon_internal_linear_layer_16_to_1(state) for i in range(4): for j in range(16): state[j] += cst[121 + i*16 + j] state[j] = state[j]**5 state = poseidon_external_linear_layer_16_to_1(state) h = F(0) for i in range(16): h += state[i] + original_state[i] return h def poseidon_linear_layer_2_to_1(state): M = Matrix(F,[[2,1],[1,2]]) return [2*state[0]+state[1],state[0]+2*state[1]] def poseidon_external_linear_layer_4_to_1(state): M_4 = [[5,7,1,3],[4,6,1,1],[1,3,5,7],[1,1,4,6]] new_state = [0 for i in range(4)] for i in range(4): for j in range(4): new_state[i] += M_4[i][j] * state[j] return new_state def poseidon_external_linear_layer_16_to_1(state): M_E = [[10,14,2,6,5,7,1,3,5,7,1,3,5,7,1,3], [ 8,12,2,2,4,6,1,1,4,6,1,1,4,6,1,1], [ 2,6,10,14,1,3,5,7,1,3,5,7,1,3,5,7], [ 2,2,8,12,1,1,4,6,1,1,4,6,1,1,4,6], [ 5,7,1,3,10,14,2,6,5,7,1,3,5,7,1,3], [ 4,6,1,1,8,12,2,2,4,6,1,1,4,6,1,1], [ 1,3,5,7,2,6,10,14,1,3,5,7,1,3,5,7], [ 1,1,4,6,2,2,8,12,1,1,4,6,1,1,4,6], [ 5,7,1,3,5,7,1,3,10,14,2,6,5,7,1,3], [ 4,6,1,1,4,6,1,1,8,12,2,2,4,6,1,1], [ 1,3,5,7,1,3,5,7,2,6,10,14,1,3,5,7], [ 1,1,4,6,1,1,4,6,2,2,8,12,1,1,4,6], [ 5,7,1,3,5,7,1,3,5,7,1,3,10,14,2,6], [ 4,6,1,1,4,6,1,1,4,6,1,1,8,12,2,2], [ 1,3,5,7,1,3,5,7,1,3,5,7,2,6,10,14], [ 1,1,4,6,1,1,4,6,1,1,4,6,2,2,8,12]] new_state = [0 for i in range(16)] for i in range(16): for j in range(16): new_state[i] += M_E[i][j] * state[j] return new_state def poseidon_internal_linear_layer_4_to_1(state): M_I = [[2,1,1,1],[1,2,1,1],[1,1,4,1],[1,1,1,8]] new_state = [0 for i in range(4)] for i in range(4): for j in range(4): new_state[i] += M_I[i][j] * state[j] return new_state def poseidon_internal_linear_layer_16_to_1(state): M_I = [[68,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [1,85,1,1,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,81,1,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,95,1,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,58,1,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,90,1,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,93,1,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,40,1,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,35,1,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,25,1,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,96,1,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,22,1,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,74,1,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,69,1], [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,53]] new_state = [0 for i in range(16)] for i in range(16): for j in range(16): new_state[i] += M_I[i][j] * state[j] return new_state def anemoi_C_2_to_1(): return [ 39, 41362478282768062297187132445775312675360473883834860695283235286481594490621, 9548818195234740988996233204400874453525674173109474205108603996010297049928, 25365440569177822667580105183435418073995888230868180942004497015015045856900, 34023498397393406644117994167986720327178154686105264833093891093045919619309, 38816051319719761886041858113129205506758421478656182868737326994635468402951, 35167418087531820804128377095512663922179887277669504047069913414630376083753, 25885868839756469722325652387535232478219821850603640827385444642154834700231, 8867588811641202981080659274007552529205713737251862066053445622305818871963, 36439756010140137556111047750162544185710881404522379792044818039722752946048, 7788624504122357216765350546787885309160020166693449889975992574536033007374, 3134147137704626983201116226440762775442116005053282329971088789984415999550, 50252287380741824818995733304361249016282047978221591906573165442023106203143, 48434698978712278012409706205559577163572452744833134361195687109159129985373, 32960510617530186159512413633821386297955642598241661044178889571655571939473, 12850897859166761094422335671106280470381427571695744605265713866647560628356, 14578036872634298798382048587794204613583128573535557156943783762854124345644, 21588109842058901916690548710649523388049643745013696896704903154857389904594, 35731638686520516424752846654442973203189295883541072759390882351699754104989, 34141830003233180772153845227433233456603143306530920011579259084215824391544, 30272543670850635882116596228256005460817517173808721139136515002908946750291 ] def anemoi_D_2_to_1(): return [ 14981678621464625851270783002338847382197300714436467949315331057125308909900, 28253420209785428420233456008091632509255652343634529984400816700490470131093, 51511939407083344002778208487678590135577660247075600880835916725469990319313, 46291121544435738125248657675097664742296276807186696922340332893747842754587, 3650460179273129580093806058710273018999560093475503119057680216309578390988, 45802223370746268123059159806400152299867771061127345631244786118574025749328, 11798621276624967315721748990709309216351696098813162382053396097866233042733, 42372918959432199162670834641599336326433006968669415662488070504036922966492, 52181371244193189669553521955614617990714056725501643636576377752669773323445, 23791984554824031672195249524658580601428376029501889159059009332107176394097, 33342520831620303764059548442834699069640109058400548818586964467754352720368, 16791548253207744974576845515705461794133799104808996134617754018912057476556, 11087343419860825311828133337767238110556416596687749174422888171911517001265, 11931207770538477937808955037363240956790374856666237106403111503668796872571, 3296943608590459582451043049934874894049468383833500962645016062634514172805, 7080580976521357573320018355401935489220216583936865937104131954142364033647, 25990144965911478244481527888046366474489820502460615136523859419965697796405, 33907313384235729375566529911940467295099705980234607934575786561097199483218, 25996950265608465541351207283024962044374873682152889814392533334239395044136, 17878892320641464292190655092475335317049416605865175118054314040434534086821, 25443622609028754422863910981890932539396181992608938932620284900889552530362 ] def anemoi_C_4_to_1(): return [ [39, 17756515227822460609684409997111995494590448775258437999344446424780281143353], [41362478282768062297187132445775312675360473883834860695283235286481594490621, 3384073892082712848969991795331397937188893616190315628722966662742467187281], [9548818195234740988996233204400874453525674173109474205108603996010297049928, 51311880822158488881090781617710146800056386303122657365679608608648067582435], [25365440569177822667580105183435418073995888230868180942004497015015045856900, 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51131682674615117766578358255722474622484771145670260043231096654077231782319] ] def anemoi_C_16_to_1(): return [ [39, 17756515227822460609684409997111995494590448775258437999344446424780281143353, 10188916128123599964772546147951904500865009616764646948187915341627970346879, 3814237141406755457246679946340702245820791055503616462386588886553626328449, 31231358838611540266091127386940316382485316827738464579249222989762089961618, 3726010289701932654130304682574596267996890432970838266711107863585526844332, 36992578177313978374320714629037014712724552282717071185860782184820525992055, 6539662723010541897260760345121608837413747021964775102659796495628351576700], [41362478282768062297187132445775312675360473883834860695283235286481594490621, 3384073892082712848969991795331397937188893616190315628722966662742467187281, 38536464596998108028197905645250196649287447208374169339784649587982292038621, 37592197675289757358471908199906415982484124338112374453435292524131427342810, 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51092856030717857607132039047789240547482897962295861318467321833280572912593, 51610208211871924557451265725733951220616079019514789132032962359833072317205, 14800890894612002638570836260269548031587506768363863797633541619652896335116, 47927023617684282491494208201013569921672642612236042045401823798666133017562] ] def poseidon_round_constant_2_to_1(): return [ 44510337639712444877093863969199054965277800588455612249278638908194748645831, 21803715039317278198490310228838761820084178670568647145430631061363562182159, 7624865858307587153533893753671854337113466346291121078558552645350285711947, 40816250157678830542785454550323790288400761867270997552332922267166370848099, 26700489303136047462599262740180012654857443933973506452655094204874268181798, 29300041198680547975810813644545348954050411371551740473502764872245855641482, 26494260871076350781917504826961109818301921647993891506179327799406892257760, 51471943067203395853539598076816386277188697473371359746626216561944728278869, 48874150250826827063647140518997592549563417409147246235831213929889330889464, 4957296567799842922524759318027693610815701909959689401077625970883603151110, 49787130886622940646628207982474849305464467960406760686521606845929813913147, 38626507234346048667761615866199783635070759234617387640403950557591257611930, 43672937506493322470130890010227422460105683953151094688032165492686807529714, 10282858088808039236495153687326481750629167926398528104702176163827531439774, 5929373583590601619353793840106929273025491048347772550388315300478156302480, 40523767159781096993564794726793344971416282562716017669035314514589021856544, 35087653160263082011551011896785452178273871331090954735353760094574180797326, 44066134544197993553720315073514236799698542066082224906667320314729128689851, 46811190561503483095087189032015959148465356044838419985597715002375968521789, 35502138198479058392251639631217384470706251578088034693745546686222031522574, 3011951966042824356793101436014075881633742606023423824609791334873649401619, 36441783079799715976603149530703751751672873737838939240893549516900746063885, 5949000965032854376013985161729805610095473216976505768565157587449663833146, 39334547265154726054631299624100840161191136653442409769156478840344483284117, 44407194440944549422962884120864337491414458688079798116475114348830479824132, 17726376508115223453307205134714318843193912409715438117420622264717671262663, 756868613004458973360577644537468651009832005811964377612175868388980341238, 3421893741771938128946389260799798658478598753415463644298734220953059738355, 13293243933107737951928282334791569607692876620282367672054893180625816893632, 11906453198605884256628058547608350794281153234160543833653104246224561572116, 41007993900563419378450318427807675773574107233531791780559911428122060668864, 48053112103639043655338341411293547635466618118313162578053483741000954697443, 32234194819559922425974652761643838211443296225838831687358769666643041225472, 48939529425812404248175324082406620877605698116805848434311747711965735603142, 23172302197508009638107441698229967178757578829167657081073715837516157038684, 26828245904250884987904133111377098838723772557669646848196202419087853870872, 36025164071480125389137319620343252251920437049927443652919962965645042660420, 42147865145919705097445974287709456827305052297675196211654971979806749888911, 20585606416170880487041307637777839232517038894653375498347934603702403525799, 1047663270527934381838445994762652910090839507177449076034186708210288801902, 49864990265274912108645272682223261996354786042911445790248422528668444967688, 20274910123179255493744356413243132767746258718293295072669857132542604950741, 39934722872842035804029775488645871956511886709858512717725127998627130523912, 33295937568441654166303959882114891655347924209941192993151592385097862772126, 39198754144978337534654702520273605486297255614756323128663775493042981926264, 48114214484211668830722398263059235215883885642960292320018016482221617479308, 20857205525756474383857323509517945359548153106777905032259140536596758842151, 20481512031474492331394869498229505122694442073123511672315331466779200648987, 5512348932066875222255592673449822544023582557729178288775446420395599163714, 14748707870289380337081091822758247948394420380976550635416667891847265434773, 20722592968207591585193709289557966995643707360519106502308701025990663556112, 30345071158541998337681526950804671230825251993252779307899796752848528236301, 37998633152333475045376762610205951441947316428701472139959038991258723083573, 25187075483245106412039082847435291293567789993240499080077480139071082713811, 19421269742609173994970218921590288196829015512476170234735579602917648585528, 44645626649350902490681022627010246390651823839290377812847048196342039743308, 38015410591674700109176981164929629504879929166582206795827935147005325179107, 15907673084411204300870039215095416489657280867726923876605554605918361454411, 4839910768263945909615643698821897421248623201436406727882801614226282796736, 11993166323725114372511567048380837525145267539902083755578961856890086640616, 13920948384274828210917386586592591296235909076917892178237942711445546791673, 890670937435713979056767019654860866935017937714294844928044822115961948695, 17228860181078068965008756660486864527552317469394243328944319614877053158985, 10077644153064320976006893555780056518106113458749153233744229760482343163199, 49940012233787551970719440300197866554675665187348390484098206972627022147562, 46156268877611784805956766593634373731938578230585206172333523828368963221701, 22953533088186447995354081903847946065608888217162100116127853233026059928601, 52274456635025394989373456728632213614198023598074828050923870917877714774032, 10763107301772445560209819564880953581415947909035345171575074311681161298071, 18925434003927090811791086956853651992753402455627802833236246367412669605539, 44640541969065703218376218137336488126193962189090743983027473909004591005110, 6325804276918590364991192431609508508777152352802958405080031341778877821773 ] def poseidon_round_constant_4_to_1(): return [ 11865901593870436687704696210307853465124332568266803587887584059192277437537, 37413344849675497106163505103761203874617077416461933389729149896951619083615, 32493856687297537788073517556470839888070933486712636845483468275561142904504, 8106572321306448561272383558012749963748358844705299406391447161298410877832, 11707331945334514286120137391947350087632085623772869951863164361695922976568, 50176353669915139758684707864014381736527453065793678083699453978150986704353, 7614231165138437703715796351400512419034157550777684039873810826440625723695, 17108745804308684637964438487237723214541645477949369557189249702220750722331, 28482709556494724328894800736802198653800073555798724636385135549439085356742, 22084621272529558534346674593668266856649195963035162420257478396107381285157, 11065046937453971018193111360820446155970823623967390073553725256595768408791, 20163386569362559253936962862374791389308839238220230985809310822791529262025, 3941572053547598429065422950522133819874343789089916995553450661687375302109, 26478286544099137612981910043301624023925078357447813396543283503388535703096, 20695873604353009531429015276900191600064214090812802713214397507580078182919, 13125722302273298866746961565666978849814465870768123336569424431819752980076, 36198064501926046106576802008779486932990788729159233810880339338049275796987, 44304460846758155168737768840994398685118878625607355725968797162919830864102, 33427295080737122973704388239917504856466726697805464799279074730831006493899, 33676819385378678616140345579679379172852136878360832374035200947147349341877, 44698579309865383047943970651850256651158236406641741596599260270592257060333, 9785548396072733556484927856146778907814788851086349315764873426432645263872, 13443943788901083053739342733252043423900693132812974385986650498262359453435, 50643729733611061821734405325096960434556494526582143101458996741611494836986, 9762302758250004682914036756566790454206292929079802149918241419991743778078, 19475373737975172049750799347581927313285945820087657933615778552974211829387, 26326676308398320579169539788392437654921007121904386041440053251920191437301, 33434337088018971767011967694326287068660679954245122268447231545106574191053, 21854837019991553667332010956652219921642250105983660108852707854862240766704, 1855873836256370364169888814967543928768577071445146807885001909257610924575, 48105724357874736702308498318239595022868478610964925776544998873327877770469, 6893757226114776013992120610353413647048965744053221939747330482283347049271, 49492933790401867565879330847407697876101917412920943837692305231776568086150, 43522959518323197786977932091172575965428037444858136419745890338308804820810, 39369220628770987071776708725487093142438968678975788148890686830600891659237, 24332002500271167754445178113059124234684848763578719088484322891936508359054, 32863475623207582419161401899951874256189430526558709698459253237664899579477, 8315390532973093090228198037973187268458339671135900692256692995983001224287, 41596164941281344945126840056611498785955123869980989807278169650894369778621, 41140021013127548285923961611241892352480288807646515822914427245468443615449, 42624333566444295089232230699974262280460377983015559040916764999567317327294, 29425841969458336716648866633284898031574592123216791821970989517602546368463, 20194832349178074328255630030474794676357522951312816945265283318496141911576, 33508834389330212986852784163678812323448884912646004437785005736522859730449, 20197489266521008707527755143868210833027985912465941510937472218208331469324, 38189796622106345878699238475711002255025750905901925248625120562682573353793, 48437331749916394313065146750618123382683254942785601073233866557242834888501, 34815884667490928168338620954175830688114531237099462583592020572423301193334, 1316079587764339149090919530288539945185249635387918305928554726824292235069, 33564480841331620167847153616337187248054503582700661803825728035418602546478, 4635634898381888421672273828316335969974599848444510316738469345444620659008, 8644506076646842294589324870931361199184791348209582052726445382015132439419, 39498585060657083972778194861599167626335350278223243726925173218749695943806, 42901602831339057007425445486193581840749112246589631302873671293308101878875, 20119933204882102974459031584507100339282292349398588923453836079377072829543, 7917862289043363038204972116125424279857433068189510615392330392863075948512, 41284417024025222157952919191031968108126105524670279472881067727309802924938, 11213861995768467857413038001306057240793870929626059934261458727946548965379, 2291742710611132809700323762125675349484016058554275673428643410085506076100, 30099159053997341705317995418169313532098300934328131162175924134794709943047, 447035513285578307783519781307142266645679652807941291454847780415896684065, 19941446202184504378547837635870560393064630187876613630546846906393007677289, 7595261399959684629699197426920893479848768772071384660164934610968891053864, 51598580281806900142260694365187051410317675046136337884836978415482902327015, 12450848281586712352554721829724230078424064515794153380314705783292880037478, 5237102499670441785007944785581992844697685968922355014280712201430686167152, 1412524057853628881005630586377727487233247150373319518286783509614859257068, 38519766408760192821848550196157518411386556623071006612683448412823634200875, 11338671486975802181674275776989710780888734229624346786700048285586675342901, 23124572501783393477231165425714476214042723292141825213493635111951207504070, 21528356110015199451243279738115385806356940590132503530639630620611521954326, 6284174238932569340060925799940162325946442751185026413727709496271066916876, 48373517651545249281510690416218268384343400250317171450103976311090286221260, 27752440147182328733098243645400559151338503658043397110598417983425635093551, 43156725395743020846958899706072234263962738024394096815001388170789961679788, 48906049704561774201639151262665470255437206145980347197443063657678740507943, 24918160465086526594937065443815615610757370328053649165018481775513828479869, 36462368786443951186110721729238677880688318912401935278190656741197184273952, 37367696075403883562827939745268661270924157618310868295845006320225084994632, 42130603320119794983803262970740129474583503116320676437504579920473229006778, 12096038367976885628335054904138821822550042039079703385879844461517464118581, 34671362090033614505367959844073659507869267381932134573786004532584171425818, 15908652423714359894720614650322760756461828514699821946843077879932200328081, 7518568119601342737128460613704294443674406422237476295695786631549469567412, 33513737101700389003254558060695049730922342329295390135821890558696123720054, 49765425774819103826723198731734445691737353182147628471479513204868044796119, 48731721046471530891684818884908827036844194399863789073273030545326532602503, 1994879948378542466338304292753049990663872919840272992167645879411261807091 ] def poseidon_round_constant_16_to_1(): return [ 39725799400017827115953999199803965513668921247606107843235739645000498452181, 42966428960558994593504354654034020585855169251008976179361555763424614464338, 1902577049757491257818576950592390026062184527103985176709404045325719879153, 44431672934524375006946320990995907220611992982569305140824479722873832750184, 39726183949244760384768131039650643342100328144953562677717530936076214603575, 47922249389084318636163655667342945193486098112364797914711366318805271081986, 1862870272947949400931550187895813583996084263112126758587127421895920740217, 4609552006052426829558648842629624826577232105300129098751251434249497379415, 35588143488957689566276930373759329909335354222484070944945535758095890233602, 31992851211763793548423275170246297274462525072830105768609534303798701174374, 10431109178659867018016774068824625279897747730577584847579124215705854088752, 1949666570245048798153069638552026706752846020986409274673577266743024227986, 26013146320492118585809324011747055383255013664810283913111923543378165512435, 24326326384498087823059984407171843358848416830861907662385192677188116053544, 22319685994625011021753350147173406654223939569203496437101532456482753075879, 50557239558368781744228704045005139069258908206255515650400270553037541504198, 8150349453804124148576142676639616213878444242674057629308915686941165448719, 48208850819924081505370182430049176104131115325038330030896851480045955495846, 23292055228237110741391983617517431423114804124284097964494568447865687504083, 41624195613852190072343927309438239744583865875755371975404916906916622520312, 35035764767249963834124347515646744980548162618519514861977709647292192162935, 4779348796643887084014079273412324900195658451616750603184150546614709227590, 33367047533960399900339953965992791362189146117008495287189183308885826514987, 47818152023063189199872106125697047204743569529570459808477570336306405682618, 42796246674763432543587686312218067656639355734975037053737259369488219415432, 10957190869792979750714342710039883244593973691851341486138757518667861871308, 2841790402482039728028394542789470351099439451234544313142331893197703007354, 16616683520944631525798957628064854348665295295102366742591213566889088018633, 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16026951337367383175610820246210183497734025720194433489132017234538604936414, 33572475268799564642745000649407837861228346301126684294541252715366099579094, 26220434981437976873498592303656146371158505657662010285451539733034399969825, 31040488125858696173362986090795905945928951823554908571888900564555975057860, 47286385036886749775224009536390346046643937618292722852399698152345906543537, 20950917282535983122464307959293663918938997314569186126689031705635737272062, 16685712499755301665281386819771726363157372904949135525550583883571962834528, 18683291445525541017294795892345078328382688847480507122165708511223740177458, 2608268839331669212463985078421319322001352992807108968314090990777368891770, 40037105172855926626375817902131326490267684706159606232991633054381953553026, 21626330967116418140505001603028197974177240574915763170077069271066169546158, 35330469786033362269122965704661462637931218147681429112041541870602933338162, 7959740499179483922969783988740981409045430979967212583073048384801003055527 ] R = RealField(500) #Real numbers with precision 500 bits if len(sys.argv) != Integer(7): print("Usage: