| /* |
| Licensed to the Apache Software Foundation (ASF) under one |
| or more contributor license agreements. See the NOTICE file |
| distributed with this work for additional information |
| regarding copyright ownership. The ASF licenses this file |
| to you under the Apache License, Version 2.0 (the |
| "License"); you may not use this file except in compliance |
| with the License. You may obtain a copy of the License at |
| |
| http://www.apache.org/licenses/LICENSE-2.0 |
| |
| Unless required by applicable law or agreed to in writing, |
| software distributed under the License is distributed on an |
| "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY |
| KIND, either express or implied. See the License for the |
| specific language governing permissions and limitations |
| under the License. |
| */ |
| |
| // |
| // pair.swift |
| // |
| // Created by Michael Scott on 07/07/2015. |
| // Copyright (c) 2015 Michael Scott. All rights reserved. |
| // |
| |
| /* AMCL BN Curve Pairing functions */ |
| |
| public struct PAIR { |
| |
| // Line function |
| static func linedbl(_ A: inout ECP2,_ Qx:FP,_ Qy:FP) -> FP12 |
| { |
| var a:FP4 |
| var b:FP4 |
| var c:FP4 |
| |
| var XX=FP2(A.getx()) //X |
| var YY=FP2(A.gety()) //Y |
| var ZZ=FP2(A.getz()) //Z |
| var YZ=FP2(YY) //Y |
| YZ.mul(ZZ) //YZ |
| XX.sqr() //X^2 |
| YY.sqr() //Y^2 |
| ZZ.sqr() //Z^2 |
| |
| YZ.imul(4) |
| YZ.neg(); YZ.norm() //-2YZ |
| YZ.pmul(Qy) //-2YZ.Ys |
| |
| XX.imul(6) //3X^2 |
| XX.pmul(Qx) //3X^2.Xs |
| |
| let sb=3*ROM.CURVE_B_I |
| ZZ.imul(sb) |
| if ECP.SEXTIC_TWIST == ECP.D_TYPE { |
| ZZ.div_ip2(); |
| } |
| if ECP.SEXTIC_TWIST == ECP.M_TYPE { |
| ZZ.mul_ip() |
| ZZ.add(ZZ) |
| ZZ.norm() |
| YZ.mul_ip() |
| YZ.norm() |
| } |
| ZZ.norm() // 3b.Z^2 |
| |
| YY.add(YY) |
| ZZ.sub(YY); ZZ.norm() // 3b.Z^2-Y^2 |
| |
| a=FP4(YZ,ZZ) // -2YZ.Ys | 3b.Z^2-Y^2 | 3X^2.Xs |
| if ECP.SEXTIC_TWIST == ECP.D_TYPE { |
| b=FP4(XX) // L(0,1) | L(0,0) | L(1,0) |
| c=FP4(0) |
| } else { |
| b=FP4(0) |
| c=FP4(XX); c.times_i() |
| } |
| A.dbl() |
| |
| return FP12(a,b,c) |
| } |
| |
| |
| // Line function |
| static func lineadd(_ A: inout ECP2,_ B:ECP2,_ Qx:FP,_ Qy:FP) -> FP12 |
| { |
| var a:FP4 |
| var b:FP4 |
| var c:FP4 |
| |
| var X1=FP2(A.getx()) // X1 |
| var Y1=FP2(A.gety()) // Y1 |
| var T1=FP2(A.getz()) // Z1 |
| var T2=FP2(A.getz()) // Z1 |
| |
| T1.mul(B.gety()) // T1=Z1.Y2 |
| T2.mul(B.getx()) // T2=Z1.X2 |
| |
| X1.sub(T2); X1.norm() // X1=X1-Z1.X2 |
| Y1.sub(T1); Y1.norm() // Y1=Y1-Z1.Y2 |
| |
| T1.copy(X1) // T1=X1-Z1.X2 |
| X1.pmul(Qy) // X1=(X1-Z1.X2).Ys |
| if ECP.SEXTIC_TWIST == ECP.M_TYPE { |
| X1.mul_ip() |
| X1.norm() |
| } |
| T1.mul(B.gety()) // T1=(X1-Z1.X2).Y2 |
| |
| T2.copy(Y1) // T2=Y1-Z1.Y2 |
| T2.mul(B.getx()) // T2=(Y1-Z1.Y2).X2 |
| T2.sub(T1); T2.norm() // T2=(Y1-Z1.Y2).X2 - (X1-Z1.X2).Y2 |
| Y1.pmul(Qx); Y1.neg(); Y1.norm() // Y1=-(Y1-Z1.Y2).Xs |
| |
| a=FP4(X1,T2) // (X1-Z1.X2).Ys | (Y1-Z1.Y2).X2 - (X1-Z1.X2).Y2 | - (Y1-Z1.Y2).Xs |
| if ECP.SEXTIC_TWIST == ECP.D_TYPE { |
| b=FP4(Y1) |
| c=FP4(0) |
| } else { |
| b=FP4(0) |
| c=FP4(Y1); c.times_i() |
| } |
| A.add(B) |
| return FP12(a,b,c) |
| } |
| |
| |
| // Optimal R-ate pairing |
| static public func ate(_ P1:ECP2,_ Q1:ECP) -> FP12 |
| { |
| var f=FP2(BIG(ROM.Fra),BIG(ROM.Frb)) |
| let x=BIG(ROM.CURVE_Bnx) |
| var n=BIG(x) |
| var K=ECP2() |
| |
| var lv:FP12 |
| |
| if ECP.CURVE_PAIRING_TYPE == ECP.BN { |
| if ECP.SEXTIC_TWIST == ECP.M_TYPE { |
| f.inverse() |
| f.norm() |
| } |
| n.pmul(6); |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| n.dec(2) |
| } else { |
| n.inc(2) |
| } |
| } else {n.copy(x)} |
| |
| n.norm() |
| |
| var n3=BIG(n) |
| n3.pmul(3) |
| n3.norm() |
| |
| var P=ECP2(); P.copy(P1); P.affine() |
| var Q=ECP(); Q.copy(Q1); Q.affine() |
| |
| |
| let Qx=FP(Q.getx()) |
| let Qy=FP(Q.gety()) |
| |
| var A=ECP2() |
| A.copy(P) |
| |
| var NP=ECP2() |
| NP.copy(P) |
| NP.neg() |
| |
| var r=FP12(1) |
| let nb=n3.nbits() |
| |
| for i in (1...nb-2).reversed() |
| //for var i=nb-2;i>=1;i-- |
| { |
| r.sqr() |
| lv=linedbl(&A,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| let bt=n3.bit(UInt(i))-n.bit(UInt(i)) |
| if bt == 1 { |
| lv=lineadd(&A,P,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| } |
| if bt == -1 { |
| //P.neg() |
| lv=lineadd(&A,NP,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| //P.neg() |
| } |
| } |
| |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| r.conj() |
| } |
| |
| // R-ate fixup required for BN curves |
| |
| if ECP.CURVE_PAIRING_TYPE == ECP.BN { |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| //r.conj() |
| A.neg() |
| } |
| K.copy(P) |
| K.frob(f) |
| |
| lv=lineadd(&A,K,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| K.frob(f) |
| K.neg() |
| lv=lineadd(&A,K,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| } |
| return r |
| } |
| // Optimal R-ate double pairing e(P,Q).e(R,S) |
| static public func ate2(_ P1:ECP2,_ Q1:ECP,_ R1:ECP2,_ S1:ECP) -> FP12 |
| { |
| var f=FP2(BIG(ROM.Fra),BIG(ROM.Frb)) |
| let x=BIG(ROM.CURVE_Bnx) |
| var n=BIG(x) |
| var K=ECP2() |
| var lv:FP12 |
| |
| if ECP.CURVE_PAIRING_TYPE == ECP.BN { |
| if ECP.SEXTIC_TWIST == ECP.M_TYPE { |
| f.inverse() |
| f.norm() |
| } |
| n.pmul(6); |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| n.dec(2) |
| } else { |
| n.inc(2) |
| } |
| } else {n.copy(x)} |
| |
| n.norm() |
| var n3=BIG(n) |
| n3.pmul(3) |
| n3.norm() |
| |
| var P=ECP2(); P.copy(P1); P.affine() |
| var Q=ECP(); Q.copy(Q1); Q.affine() |
| var R=ECP2(); R.copy(R1); R.affine() |
| var S=ECP(); S.copy(S1); S.affine() |
| |
| |
| let Qx=FP(Q.getx()) |
| let Qy=FP(Q.gety()) |
| let Sx=FP(S.getx()) |
| let Sy=FP(S.gety()) |
| |
| var A=ECP2() |
| var B=ECP2() |
| var r=FP12(1) |
| |
| A.copy(P) |
| B.copy(R) |
| var NP=ECP2() |
| NP.copy(P) |
| NP.neg() |
| var NR=ECP2() |
| NR.copy(R) |
| NR.neg() |
| |
| |
| let nb=n3.nbits() |
| |
| for i in (1...nb-2).reversed() |
| //for var i=nb-2;i>=1;i-- |
| { |
| r.sqr() |
| lv=linedbl(&A,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| lv=linedbl(&B,Sx,Sy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| let bt=n3.bit(UInt(i))-n.bit(UInt(i)) |
| |
| if bt == 1 { |
| lv=lineadd(&A,P,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| lv=lineadd(&B,R,Sx,Sy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| } |
| |
| if bt == -1 { |
| //P.neg(); |
| lv=lineadd(&A,NP,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| //P.neg(); |
| //R.neg() |
| lv=lineadd(&B,NR,Sx,Sy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| //R.neg() |
| } |
| |
| } |
| |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| r.conj() |
| } |
| |
| // R-ate fixup required for BN curves |
| |
| if ECP.CURVE_PAIRING_TYPE == ECP.BN { |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| //r.conj() |
| A.neg() |
| B.neg() |
| } |
| K.copy(P) |
| K.frob(f) |
| |
| lv=lineadd(&A,K,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| K.frob(f) |
| K.neg() |
| lv=lineadd(&A,K,Qx,Qy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| |
| K.copy(R) |
| K.frob(f) |
| |
| lv=lineadd(&B,K,Sx,Sy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| K.frob(f) |
| K.neg() |
| lv=lineadd(&B,K,Sx,Sy) |
| r.smul(lv,ECP.SEXTIC_TWIST) |
| } |
| return r |
| } |
| |
| // final exponentiation - keep separate for multi-pairings and to avoid thrashing stack |
| static public func fexp(_ m:FP12) -> FP12 |
| { |
| let f=FP2(BIG(ROM.Fra),BIG(ROM.Frb)); |
| let x=BIG(ROM.CURVE_Bnx) |
| var r=FP12(m) |
| |
| // Easy part of final exp |
| var lv=FP12(r) |
| lv.inverse() |
| r.conj() |
| |
| r.mul(lv) |
| lv.copy(r) |
| r.frob(f) |
| r.frob(f) |
| r.mul(lv) |
| |
| // Hard part of final exp |
| if ECP.CURVE_PAIRING_TYPE == ECP.BN { |
| lv.copy(r) |
| lv.frob(f) |
| var x0=FP12(lv) |
| x0.frob(f) |
| lv.mul(r) |
| x0.mul(lv) |
| x0.frob(f) |
| var x1=FP12(r) |
| x1.conj() |
| var x4=r.pow(x) |
| if ECP.SIGN_OF_X == ECP.POSITIVEX { |
| x4.conj() |
| } |
| var x3=FP12(x4) |
| x3.frob(f) |
| |
| var x2=x4.pow(x) |
| if ECP.SIGN_OF_X == ECP.POSITIVEX { |
| x2.conj() |
| } |
| var x5=FP12(x2); x5.conj() |
| lv=x2.pow(x) |
| if ECP.SIGN_OF_X == ECP.POSITIVEX { |
| lv.conj() |
| } |
| x2.frob(f) |
| r.copy(x2); r.conj() |
| |
| x4.mul(r) |
| x2.frob(f) |
| |
| r.copy(lv) |
| r.frob(f) |
| lv.mul(r) |
| |
| lv.usqr() |
| lv.mul(x4) |
| lv.mul(x5) |
| r.copy(x3) |
| r.mul(x5) |
| r.mul(lv) |
| lv.mul(x2) |
| r.usqr() |
| r.mul(lv) |
| r.usqr() |
| lv.copy(r) |
| lv.mul(x1) |
| r.mul(x0) |
| lv.usqr() |
| r.mul(lv) |
| r.reduce() |
| } else { |
| var x0=FP12(r) |
| var x1=FP12(r) |
| lv.copy(r); lv.frob(f) |
| var x3=FP12(lv); x3.conj(); x1.mul(x3) |
| lv.frob(f); lv.frob(f) |
| x1.mul(lv) |
| |
| r.copy(r.pow(x)) //r=r.pow(x); |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| r.conj() |
| } |
| x3.copy(r); x3.conj(); x1.mul(x3) |
| lv.copy(r); lv.frob(f) |
| x0.mul(lv) |
| lv.frob(f) |
| x1.mul(lv) |
| lv.frob(f) |
| x3.copy(lv); x3.conj(); x0.mul(x3) |
| |
| r.copy(r.pow(x)) |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| r.conj() |
| } |
| x0.mul(r) |
| lv.copy(r); lv.frob(f); lv.frob(f) |
| x3.copy(lv); x3.conj(); x0.mul(x3) |
| lv.frob(f) |
| x1.mul(lv) |
| |
| r.copy(r.pow(x)) |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| r.conj() |
| } |
| lv.copy(r); lv.frob(f) |
| x3.copy(lv); x3.conj(); x0.mul(x3) |
| lv.frob(f) |
| x1.mul(lv) |
| |
| r.copy(r.pow(x)) |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| r.conj() |
| } |
| x3.copy(r); x3.conj(); x0.mul(x3) |
| lv.copy(r); lv.frob(f) |
| x1.mul(lv) |
| |
| r.copy(r.pow(x)) |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| r.conj() |
| } |
| x1.mul(r) |
| |
| x0.usqr() |
| x0.mul(x1) |
| r.copy(x0) |
| r.reduce() |
| } |
| return r |
| } |
| |
| // GLV method |
| static func glv(_ e:BIG) -> [BIG] |
| { |
| var u=[BIG](); |
| if ECP.CURVE_PAIRING_TYPE == ECP.BN { |
| var t=BIG(0) |
| let q=BIG(ROM.CURVE_Order) |
| var v=[BIG](); |
| for _ in 0 ..< 2 |
| { |
| u.append(BIG(0)) |
| v.append(BIG(0)) |
| } |
| |
| for i in 0 ..< 2 |
| { |
| t.copy(BIG(ROM.CURVE_W[i])) |
| var d=BIG.mul(t,e) |
| v[i].copy(d.div(q)) |
| } |
| u[0].copy(e); |
| for i in 0 ..< 2 |
| { |
| for j in 0 ..< 2 |
| { |
| t.copy(BIG(ROM.CURVE_SB[j][i])) |
| t.copy(BIG.modmul(v[j],t,q)) |
| u[i].add(q) |
| u[i].sub(t) |
| u[i].mod(q) |
| } |
| } |
| } else { // -(x^2).P = (Beta.x,y) |
| let q=BIG(ROM.CURVE_Order) |
| let x=BIG(ROM.CURVE_Bnx) |
| let x2=BIG.smul(x,x) |
| u.append(BIG(e)) |
| u[0].mod(x2) |
| u.append(BIG(e)) |
| u[1].div(x2) |
| u[1].rsub(q) |
| |
| } |
| return u |
| } |
| // Galbraith & Scott Method |
| static func gs(_ e:BIG) -> [BIG] |
| { |
| var u=[BIG](); |
| if ECP.CURVE_PAIRING_TYPE == ECP.BN { |
| var t=BIG(0) |
| let q=BIG(ROM.CURVE_Order) |
| var v=[BIG](); |
| for _ in 0 ..< 4 |
| { |
| u.append(BIG(0)) |
| v.append(BIG(0)) |
| } |
| |
| for i in 0 ..< 4 |
| { |
| t.copy(BIG(ROM.CURVE_WB[i])) |
| var d=BIG.mul(t,e) |
| v[i].copy(d.div(q)) |
| } |
| u[0].copy(e); |
| for i in 0 ..< 4 |
| { |
| for j in 0 ..< 4 |
| { |
| t.copy(BIG(ROM.CURVE_BB[j][i])) |
| t.copy(BIG.modmul(v[j],t,q)) |
| u[i].add(q) |
| u[i].sub(t) |
| u[i].mod(q) |
| } |
| |
| } |
| } else { |
| let q=BIG(ROM.CURVE_Order) |
| let x=BIG(ROM.CURVE_Bnx) |
| var w=BIG(e) |
| for i in 0 ..< 3 |
| { |
| u.append(BIG(w)) |
| u[i].mod(x) |
| w.div(x) |
| } |
| u.append(BIG(w)) |
| if ECP.SIGN_OF_X == ECP.NEGATIVEX { |
| u[1].copy(BIG.modneg(u[1],q)) |
| u[3].copy(BIG.modneg(u[3],q)) |
| } |
| } |
| return u |
| } |
| |
| // Multiply P by e in group G1 |
| static public func G1mul(_ P:ECP,_ e:BIG) -> ECP |
| { |
| var R:ECP |
| if (ROM.USE_GLV) |
| { |
| //P.affine() |
| R=ECP() |
| R.copy(P) |
| var Q=ECP() |
| Q.copy(P); Q.affine() |
| let q=BIG(ROM.CURVE_Order) |
| let cru=FP(BIG(ROM.CURVE_Cru)) |
| var t=BIG(0) |
| var u=PAIR.glv(e) |
| Q.mulx(cru); |
| |
| var np=u[0].nbits() |
| t.copy(BIG.modneg(u[0],q)) |
| var nn=t.nbits() |
| if (nn<np) |
| { |
| u[0].copy(t) |
| R.neg() |
| } |
| |
| np=u[1].nbits() |
| t.copy(BIG.modneg(u[1],q)) |
| nn=t.nbits() |
| if (nn<np) |
| { |
| u[1].copy(t) |
| Q.neg() |
| } |
| u[0].norm() |
| u[1].norm() |
| R=R.mul2(u[0],Q,u[1]) |
| } |
| else |
| { |
| R=P.mul(e) |
| } |
| return R |
| } |
| |
| // Multiply P by e in group G2 |
| static public func G2mul(_ P:ECP2,_ e:BIG) -> ECP2 |
| { |
| var R:ECP2 |
| if (ROM.USE_GS_G2) |
| { |
| var Q=[ECP2]() |
| var f=FP2(BIG(ROM.Fra),BIG(ROM.Frb)); |
| let q=BIG(ROM.CURVE_Order); |
| var u=PAIR.gs(e); |
| |
| if ECP.SEXTIC_TWIST == ECP.M_TYPE { |
| f.inverse() |
| f.norm() |
| } |
| |
| var t=BIG(0) |
| //P.affine() |
| Q.append(ECP2()) |
| Q[0].copy(P); |
| for i in 1 ..< 4 |
| { |
| Q.append(ECP2()); Q[i].copy(Q[i-1]); |
| Q[i].frob(f); |
| } |
| for i in 0 ..< 4 |
| { |
| let np=u[i].nbits() |
| t.copy(BIG.modneg(u[i],q)) |
| let nn=t.nbits() |
| if (nn<np) |
| { |
| u[i].copy(t) |
| Q[i].neg() |
| } |
| u[i].norm() |
| } |
| |
| R=ECP2.mul4(Q,u) |
| } |
| else |
| { |
| R=P.mul(e) |
| } |
| return R; |
| } |
| // f=f^e |
| // Note that this method requires a lot of RAM! Better to use compressed XTR method, see FP4.java |
| static public func GTpow(_ d:FP12,_ e:BIG) -> FP12 |
| { |
| var r:FP12 |
| if (ROM.USE_GS_GT) |
| { |
| var g=[FP12]() |
| let f=FP2(BIG(ROM.Fra),BIG(ROM.Frb)) |
| let q=BIG(ROM.CURVE_Order) |
| var t=BIG(0) |
| |
| var u=gs(e) |
| g.append(FP12(0)) |
| g[0].copy(d); |
| for i in 1 ..< 4 |
| { |
| g.append(FP12(0)); g[i].copy(g[i-1]) |
| g[i].frob(f) |
| } |
| for i in 0 ..< 4 |
| { |
| let np=u[i].nbits() |
| t.copy(BIG.modneg(u[i],q)) |
| let nn=t.nbits() |
| if (nn<np) |
| { |
| u[i].copy(t) |
| g[i].conj() |
| } |
| u[i].norm() |
| } |
| r=FP12.pow4(g,u) |
| } |
| else |
| { |
| r=d.pow(e) |
| } |
| return r |
| } |
| // test group membership - no longer needed |
| // with GT-Strong curve, now only check that m!=1, conj(m)*m==1, and m.m^{p^4}=m^{p^2} |
| /* |
| static func GTmember(m:FP12) -> Bool |
| { |
| if m.isunity() {return false} |
| let r=FP12(m) |
| r.conj() |
| r.mul(m) |
| if !r.isunity() {return false} |
| |
| let f=FP2(BIG(ROM.Fra),BIG(ROM.Frb)) |
| |
| r.copy(m); r.frob(f); r.frob(f) |
| var w=FP12(r); w.frob(f); w.frob(f) |
| w.mul(m) |
| if !ROM.GT_STRONG |
| { |
| if !w.equals(r) {return false} |
| let x=BIG(ROM.CURVE_Bnx) |
| r.copy(m); w=r.pow(x); w=w.pow(x) |
| r.copy(w); r.sqr(); r.mul(w); r.sqr() |
| w.copy(m); w.frob(f) |
| } |
| return w.equals(r) |
| } |
| */ |
| } |
| |
