| /* |
| 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. |
| */ |
| |
| /* AMCL BN Curve Pairing functions */ |
| |
| package amcl |
| |
| //import "fmt" |
| |
| /* Line function */ |
| func line(A *ECP2, B *ECP2, Qx *FP, Qy *FP) *FP12 { |
| P := NewECP2() |
| |
| P.copy(A) |
| ZZ := NewFP2copy(P.getz()) |
| ZZ.sqr() |
| var D int |
| if A == B { |
| D = A.dbl() |
| } else { |
| D = A.add(B) |
| } |
| |
| if D < 0 { |
| return NewFP12int(1) |
| } |
| |
| Z3 := NewFP2copy(A.getz()) |
| |
| var a *FP4 |
| var b *FP4 |
| c := NewFP4int(0) |
| |
| if D == 0 { /* Addition */ |
| X := NewFP2copy(B.getx()) |
| Y := NewFP2copy(B.gety()) |
| T := NewFP2copy(P.getz()) |
| T.mul(Y) |
| ZZ.mul(T) |
| |
| NY := NewFP2copy(P.gety()) |
| NY.neg() |
| ZZ.add(NY) |
| Z3.pmul(Qy) |
| T.mul(P.getx()) |
| X.mul(NY) |
| T.add(X) |
| a = NewFP4fp2s(Z3, T) |
| ZZ.neg() |
| ZZ.pmul(Qx) |
| b = NewFP4fp2(ZZ) |
| } else { /* Doubling */ |
| X := NewFP2copy(P.getx()) |
| Y := NewFP2copy(P.gety()) |
| T := NewFP2copy(P.getx()) |
| T.sqr() |
| T.imul(3) |
| |
| Y.sqr() |
| Y.add(Y) |
| Z3.mul(ZZ) |
| Z3.pmul(Qy) |
| |
| X.mul(T) |
| X.sub(Y) |
| a = NewFP4fp2s(Z3, X) |
| T.neg() |
| ZZ.mul(T) |
| ZZ.pmul(Qx) |
| b = NewFP4fp2(ZZ) |
| } |
| return NewFP12fp4s(a, b, c) |
| } |
| |
| /* Optimal R-ate pairing */ |
| func ate(P *ECP2, Q *ECP) *FP12 { |
| f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb)) |
| x := NewBIGints(CURVE_Bnx) |
| n := NewBIGcopy(x) |
| K := NewECP2() |
| var lv *FP12 |
| n.pmul(6) |
| n.dec(2) |
| n.norm() |
| P.affine() |
| Q.affine() |
| Qx := NewFPcopy(Q.getx()) |
| Qy := NewFPcopy(Q.gety()) |
| |
| A := NewECP2() |
| r := NewFP12int(1) |
| |
| A.copy(P) |
| nb := n.nbits() |
| |
| for i := nb - 2; i >= 1; i-- { |
| lv = line(A, A, Qx, Qy) |
| r.smul(lv) |
| |
| if n.bit(i) == 1 { |
| lv = line(A, P, Qx, Qy) |
| r.smul(lv) |
| } |
| r.sqr() |
| } |
| |
| lv = line(A, A, Qx, Qy) |
| r.smul(lv) |
| |
| /* R-ate fixup */ |
| |
| r.conj() |
| |
| K.copy(P) |
| K.frob(f) |
| A.neg() |
| lv = line(A, K, Qx, Qy) |
| r.smul(lv) |
| K.frob(f) |
| K.neg() |
| lv = line(A, K, Qx, Qy) |
| r.smul(lv) |
| |
| return r |
| } |
| |
| /* Optimal R-ate double pairing e(P,Q).e(R,S) */ |
| func ate2(P *ECP2, Q *ECP, R *ECP2, S *ECP) *FP12 { |
| f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb)) |
| x := NewBIGints(CURVE_Bnx) |
| n := NewBIGcopy(x) |
| K := NewECP2() |
| var lv *FP12 |
| n.pmul(6) |
| n.dec(2) |
| n.norm() |
| P.affine() |
| Q.affine() |
| R.affine() |
| S.affine() |
| |
| Qx := NewFPcopy(Q.getx()) |
| Qy := NewFPcopy(Q.gety()) |
| Sx := NewFPcopy(S.getx()) |
| Sy := NewFPcopy(S.gety()) |
| |
| A := NewECP2() |
| B := NewECP2() |
| r := NewFP12int(1) |
| |
| A.copy(P) |
| B.copy(R) |
| nb := n.nbits() |
| |
| for i := nb - 2; i >= 1; i-- { |
| lv = line(A, A, Qx, Qy) |
| r.smul(lv) |
| lv = line(B, B, Sx, Sy) |
| r.smul(lv) |
| |
| if n.bit(i) == 1 { |
| lv = line(A, P, Qx, Qy) |
| r.smul(lv) |
| lv = line(B, R, Sx, Sy) |
| r.smul(lv) |
| } |
| r.sqr() |
| } |
| |
| lv = line(A, A, Qx, Qy) |
| r.smul(lv) |
| |
| lv = line(B, B, Sx, Sy) |
| r.smul(lv) |
| |
| /* R-ate fixup */ |
| r.conj() |
| |
| K.copy(P) |
| K.frob(f) |
| A.neg() |
| lv = line(A, K, Qx, Qy) |
| r.smul(lv) |
| K.frob(f) |
| K.neg() |
| lv = line(A, K, Qx, Qy) |
| r.smul(lv) |
| |
| K.copy(R) |
| K.frob(f) |
| B.neg() |
| lv = line(B, K, Sx, Sy) |
| r.smul(lv) |
| K.frob(f) |
| K.neg() |
| lv = line(B, K, Sx, Sy) |
| r.smul(lv) |
| |
| return r |
| } |
| |
| /* final exponentiation - keep separate for multi-pairings and to avoid thrashing stack */ |
| func fexp(m *FP12) *FP12 { |
| f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb)) |
| x := NewBIGints(CURVE_Bnx) |
| r := NewFP12copy(m) |
| |
| /* Easy part of final exp */ |
| lv := NewFP12copy(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 */ |
| lv.copy(r) |
| lv.frob(f) |
| x0 := NewFP12copy(lv) |
| x0.frob(f) |
| lv.mul(r) |
| x0.mul(lv) |
| x0.frob(f) |
| x1 := NewFP12copy(r) |
| x1.conj() |
| x4 := r.pow(x) |
| |
| x3 := NewFP12copy(x4) |
| x3.frob(f) |
| |
| x2 := x4.pow(x) |
| |
| x5 := NewFP12copy(x2) |
| x5.conj() |
| lv = x2.pow(x) |
| |
| 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() |
| return r |
| } |
| |
| /* GLV method */ |
| func glv(e *BIG) []*BIG { |
| t := NewBIGint(0) |
| q := NewBIGints(CURVE_Order) |
| var u []*BIG |
| var v []*BIG |
| |
| for i := 0; i < 2; i++ { |
| t.copy(NewBIGints(CURVE_W[i])) // why not just t=new BIG(ROM.CURVE_W[i]); |
| d := mul(t, e) |
| v = append(v, NewBIGcopy(d.div(q))) |
| u = append(u, NewBIGint(0)) |
| } |
| u[0].copy(e) |
| for i := 0; i < 2; i++ { |
| for j := 0; j < 2; j++ { |
| t.copy(NewBIGints(CURVE_SB[j][i])) |
| t.copy(modmul(v[j], t, q)) |
| u[i].add(q) |
| u[i].sub(t) |
| u[i].mod(q) |
| } |
| } |
| return u |
| } |
| |
| /* Galbraith & Scott Method */ |
| func gs(e *BIG) []*BIG { |
| t := NewBIGint(0) |
| q := NewBIGints(CURVE_Order) |
| var u []*BIG |
| var v []*BIG |
| for i := 0; i < 4; i++ { |
| t.copy(NewBIGints(CURVE_WB[i])) |
| d := mul(t, e) |
| v = append(v, NewBIGcopy(d.div(q))) |
| u = append(u, NewBIGint(0)) |
| } |
| u[0].copy(e) |
| for i := 0; i < 4; i++ { |
| for j := 0; j < 4; j++ { |
| t.copy(NewBIGints(CURVE_BB[j][i])) |
| t.copy(modmul(v[j], t, q)) |
| u[i].add(q) |
| u[i].sub(t) |
| u[i].mod(q) |
| } |
| } |
| return u |
| } |
| |
| /* Multiply P by e in group G1 */ |
| func G1mul(P *ECP, e *BIG) *ECP { |
| var R *ECP |
| if USE_GLV { |
| P.affine() |
| R = NewECP() |
| R.copy(P) |
| Q := NewECP() |
| Q.copy(P) |
| q := NewBIGints(CURVE_Order) |
| cru := NewFPbig(NewBIGints(CURVE_Cru)) |
| t := NewBIGint(0) |
| u := glv(e) |
| Q.getx().mul(cru) |
| |
| np := u[0].nbits() |
| t.copy(modneg(u[0], q)) |
| nn := t.nbits() |
| if nn < np { |
| u[0].copy(t) |
| R.neg() |
| } |
| |
| np = u[1].nbits() |
| t.copy(modneg(u[1], q)) |
| nn = t.nbits() |
| if nn < np { |
| u[1].copy(t) |
| Q.neg() |
| } |
| |
| R = R.mul2(u[0], Q, u[1]) |
| |
| } else { |
| R = P.mul(e) |
| } |
| return R |
| } |
| |
| /* Multiply P by e in group G2 */ |
| func G2mul(P *ECP2, e *BIG) *ECP2 { |
| var R *ECP2 |
| if USE_GS_G2 { |
| var Q []*ECP2 |
| f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb)) |
| q := NewBIGints(CURVE_Order) |
| u := gs(e) |
| |
| t := NewBIGint(0) |
| P.affine() |
| Q = append(Q, NewECP2()) |
| Q[0].copy(P) |
| for i := 1; i < 4; i++ { |
| Q = append(Q, NewECP2()) |
| Q[i].copy(Q[i-1]) |
| Q[i].frob(f) |
| } |
| for i := 0; i < 4; i++ { |
| np := u[i].nbits() |
| t.copy(modneg(u[i], q)) |
| nn := t.nbits() |
| if nn < np { |
| u[i].copy(t) |
| Q[i].neg() |
| } |
| } |
| R = 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 */ |
| func GTpow(d *FP12, e *BIG) *FP12 { |
| var r *FP12 |
| if USE_GS_GT { |
| var g []*FP12 |
| f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb)) |
| q := NewBIGints(CURVE_Order) |
| t := NewBIGint(0) |
| |
| u := gs(e) |
| |
| g = append(g, NewFP12copy(d)) |
| for i := 1; i < 4; i++ { |
| g = append(g, NewFP12int(0)) |
| g[i].copy(g[i-1]) |
| g[i].frob(f) |
| } |
| for i := 0; i < 4; i++ { |
| np := u[i].nbits() |
| t.copy(modneg(u[i], q)) |
| nn := t.nbits() |
| if nn < np { |
| u[i].copy(t) |
| g[i].conj() |
| } |
| } |
| r = pow4(g, u) |
| } else { |
| r = d.pow(e) |
| } |
| return r |
| } |
| |
| /* test group membership */ |
| /* with GT-Strong curve, now only check that m!=1, conj(m)*m==1, and m.m^{p^4}=m^{p^2} */ |
| func GTmember(m *FP12) bool { |
| if m.isunity() { |
| return false |
| } |
| r := NewFP12copy(m) |
| r.conj() |
| r.mul(m) |
| if !r.isunity() { |
| return false |
| } |
| |
| f := NewFP2bigs(NewBIGints(CURVE_Fra), NewBIGints(CURVE_Frb)) |
| |
| r.copy(m) |
| r.frob(f) |
| r.frob(f) |
| w := NewFP12copy(r) |
| w.frob(f) |
| w.frob(f) |
| w.mul(m) |
| if !GT_STRONG { |
| if !w.equals(r) { |
| return false |
| } |
| x := NewBIGints(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) |
| } |
| |
| /* |
| func main() { |
| |
| Q:=NewECPbigs(NewBIGints(CURVE_Gx),NewBIGints(CURVE_Gy)) |
| P:=NewECP2fp2s(NewFP2bigs(NewBIGints(CURVE_Pxa),NewBIGints(CURVE_Pxb)),NewFP2bigs(NewBIGints(CURVE_Pya),NewBIGints(CURVE_Pyb))) |
| |
| r:=NewBIGints(CURVE_Order) |
| xa:=NewBIGints(CURVE_Pxa) |
| |
| fmt.Printf("P= "+P.toString()) |
| fmt.Printf("\n"); |
| fmt.Printf("Q= "+Q.toString()); |
| fmt.Printf("\n"); |
| |
| //m:=NewBIGint(17) |
| |
| e:=ate(P,Q) |
| fmt.Printf("\ne= "+e.toString()) |
| fmt.Printf("\n") |
| |
| e=fexp(e) |
| // e=GTpow(e,m); |
| |
| fmt.Printf("\ne= "+e.toString()) |
| fmt.Printf("\n"); |
| GLV:=glv(r) |
| |
| fmt.Printf("GLV[0]= "+GLV[0].toString()) |
| fmt.Printf("\n") |
| |
| fmt.Printf("GLV[0]= "+GLV[1].toString()) |
| fmt.Printf("\n") |
| |
| G:=NewECP(); G.copy(Q) |
| R:=NewECP2(); R.copy(P) |
| |
| |
| e=ate(R,Q) |
| e=fexp(e) |
| |
| e=GTpow(e,xa) |
| fmt.Printf("\ne= "+e.toString()); |
| fmt.Printf("\n") |
| |
| R=G2mul(R,xa) |
| e=ate(R,G) |
| e=fexp(e) |
| |
| fmt.Printf("\ne= "+e.toString()) |
| fmt.Printf("\n") |
| |
| G=G1mul(G,xa) |
| e=ate(P,G) |
| e=fexp(e) |
| fmt.Printf("\ne= "+e.toString()) |
| fmt.Printf("\n") |
| } |
| */ |