Koblitz Miller field Fp p prime field Fp E Fp Fp Hasse Weil 2.2 Fp 2 P Q R R P Q O P O R Q Q O R P P xp, yp Q xq, yq yp yq R=O
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1 An Internet Vote Using the Elliptic Curve Cryptosystem TAKABAYASHI Shigeki Nowadays various changes are taking place in the society by the spread of the Internet, and we will vote by the Internet using the encryption in the elections. Even if the key of the Elliptic Curve Cryptosystem is shorter than the key of the RSA cryptosystem, the Elliptic Curve Cryptosystem is secure. This paper shows the method of Internet vote with the Elliptic Curve Cryptosystem. Furthermore, the simulation of an Internet vote with the Elliptic Curve Cryptosystem is shown using Excel and VBA. 1 1 RSA RSA RSA Excel VBA
2 Koblitz Miller field Fp p prime field Fp E Fp Fp Hasse Weil 2.2 Fp 2 P Q R R P Q O P O R Q Q O R P P xp, yp Q xq, yq yp yq R=O P Q Q P yp yq R xr, yr xr yr λ xp xq
3 xp xq S d T d S S T T d S d ECElGamal 2 Fp E P r1 e r1 P r1 d e r2 R r2 P e K r2 e M C M K R, C R, C R d K d R K M C K M a Fp E1 P1
4 b r11 e1 r11 P1 r11 d1 e1 c r12 R1 r12 P1 d e1 K1 r12 e1 e M1 C1 M1 K1 a Fp E2 P2 b r21 e2 r21 P2 r21 d2 e2 c r22 R2 r22 P2 d e2 K2 r22 e2 e M2 C1 R1 C2 M2 K2 R21 R1 K2 C21 C1 K2 f R2, C2, R21, C21 a R2 d2 K 2 d2 R2 K2 b M 2 C2 K 2 M2 R 1 R21 K 2 C 1 C21 K 2 R1 C1 a M 2 C 1 b R 1 d1 K 1 d1 R 1 K 1 c M 1 C 1 K 1 M
5 Fp 300 = =17 =99 =349 a, b, p Sub yuuriten() Dim M(30000), N(30000) As Long Dim ia, ib, ih, ih1, i, j, ip, ir, iw As Long Dim wk As Worksheet Set wk = ActiveSheet Range( P2:R30000 ).Select Selection.ClearContents ia = wk.cells(5, 3).Value ib = wk.cells(6, 3).Value ih = wk.cells(7, 3).Value ih1 = ih 1 For i = 0 To ih1 M(i) = (i * i) Mod ih iw = (M(i) * i) Mod ih N(i) = (iw + ((ia * i) Mod ih) + ib) Mod ih Next i Application.ScreenUpdating = False ip = 1 For i = 0 To ih1
6 For j = 0 To ih1 If N(i) <> M(j) Then GoTo Label1 ip = ip + 1 wk.cells(ip, 16).Value = ip 1 wk.cells(ip, 17).Value = i wk.cells(ip, 18).Value = j Label1: Next j Next i wk.cells(11, 3).Value = ip Application.ScreenUpdating = True End Sub F317 1 P x y x y
7 a Fp E1 P1 b r11 e1 r11 P1 r11 1 d1 e1 1
8 a Fp E2 P2 b r21 e2 r21 P2 r21 2 d2 e a 1 b 1 a 349 b r12 R1 r12 P1 c 1 e1 K1 r12 e1
9 d M1 C1 M1 K1 a b r22 R2 r22 P2 c 2 e2 K2 r22 e2 d M2 C1 R1 C2 M2 K2 R21 R1 K2 C21 C1 K2 2 Function tasi (a, b As Long) As Long Dimia,ib,ih,i,j,ip,ir,iw,ix,iy,xa,xb,ya,yb,xc,yc,lamAsLong Dim wk As Worksheet Set wk = ActiveSheet ia = wk.cells(5, 3).Value ib = wk.cells(6, 3).Value ih = wk.cells(7, 3).Value xa=wk.cells(a+1,17).value ya=wk.cells(a+1,18).value xb=wk.cells(b+1,17).value yb=wk.cells(b+1,18).value ir = wk.cells(11, 3).Value
10 Ifa=0Orb=0Then tasi=a+b GoTo Label4 End If If (xa = xb) And (ya = (ih yb) Or ((ya + yb) = 0)) Then tasi = 0 GoTo Label4 End If If xa = xb Then GoTo Label1 ix = ya yb If ix < 0 Then ix = ix + ih iy = xa xb If iy < 0 Then iy = iy + ih lam = wari(ix, iy) GoTo Label2 Label1: ix = (3 * xa * xa + ia) Mod ih iy = (2 * ya) Mod ih lam = wari(ix, iy) Label2: xc = lam * lam xa xb xc = xc + 2 * ih xc = xc Mod ih yc = xa xc If yc < 0 Then yc = yc + ih yc = lam * yc ya If yc < 0 Then yc = yc + ih yc = yc Mod ih For i = 1 To ir If wk.cells(i, 17).Value = xc And wk.cells(i, 18).Value = yc Then GoTo Label3 Next i tasi = 99 GoTo Label4 Label3: tasi=i 1 Label4:
11 End Function R2, C2, R21, C a R2 2 d2
12 K 2 d2 R2 d2 r22 P2 r22 e2 K2 K2 b M 2 C2 K 2 M2 R 1 R21 K 2 C 1 C21 K 2 R1 C1 a M 2 C 1 b R 1 1 d1 K 1 d1 R 1 d1 r12 P1 r12 e1 K1 K1 c M 1 C 1 K 1 M1 M1 4 RSA RSA RSA RSA
13 RSA RSA researchmap RSA NTT DATA DIGITAL GOVERNMENT & FINANCIAL TOPICS public.nttdata.co.jp/topics_detail2/contents_type=9&id=923
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