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Connecting Legendre with Kummer and Edwards

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  • Scalar multiplication on suitable Legendre form elliptic curves can be speeded up in two ways. One can perform the bulk of the computation either on the associated Kummer line or on an appropriate twisted Edwards form elliptic curve. This paper provides details of moving to and from between Legendre form elliptic curves and associated Kummer line and moving to and from between Legendre form elliptic curves and related twisted Edwards form elliptic curves. Further, concrete twisted Edwards form elliptic curves are identified which correspond to known Kummer lines at the 128-bit security level which provide very fast scalar multiplication on modern architectures supporting SIMD operations.

    Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.


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  • Table 1.  Double and differential addition in the square-only setting

    $\mathsf{dbl}(\mathsf{x}^2,\mathsf{z}^2)$ $\mathsf{diffAdd}(\mathsf{x}_1^2,\mathsf{z}_1^2,\mathsf{x}_2^2,\mathsf{z}_2^2,\mathsf{x}^2,\mathsf{z}^2)$
        $\mathsf{s}_0 = \mathsf{B}^2(\mathsf{x}^2+\mathsf{z}^2)^2$;     $\mathsf{s}_0 = \mathsf{B}^2(\mathsf{x}_1^2+\mathsf{z}_1^2)(\mathsf{x}_2^2+\mathsf{z}_2^2)$;
        $\mathsf{t}_0 = \mathsf{A}^2(\mathsf{x}^2-\mathsf{z}^2)^2$;     $\mathsf{t}_0 = \mathsf{A}^2(\mathsf{x}_1^2-\mathsf{z}_1^2)(\mathsf{x}_2^2-\mathsf{z}_2^2)$;
        $\mathsf{x}_3^2 = \mathsf{b}^2(\mathsf{s}_0+\mathsf{t}_0)^2$;     $\mathsf{x}_3^2 = \mathsf{z}^2(\mathsf{s}_0+\mathsf{t}_0)^2$;
        $\mathsf{z}_3^2 = \mathsf{a}^2(\mathsf{s}_0-\mathsf{t}_0)^2$;     $\mathsf{z}_3^2 = \mathsf{x}^2(\mathsf{s}_0-\mathsf{t}_0)^2$;
        return $(\mathsf{x}_3^2,\mathsf{z}_3^2)$.     return $(\mathsf{x}_3^2,\mathsf{z}_3^2)$.
     | Show Table
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    Table 2.  Scalar multiplication on Kummer line using a ladder

    $\mathsf{scalarMult}(\mathsf{P},n)$ $\mathsf{ladder}(\mathsf{R},\mathsf{S},\mathfrak{b})$
    input: $\mathsf{P}\in{\mathcal K}_{\mathsf{a}^2,\mathsf{b}^2}$;    if ($\mathfrak{b}=0$)
          $\ell$-bit scalar $n=(1,n_{\ell-2},\ldots,n_0)$;       $\mathsf{S}=\mathsf{diffAdd}(\mathsf{R},\mathsf{S},\mathsf{P})$;
    output: $n\mathsf{P}$;          $\mathsf{R}=\mathsf{dbl}(\mathsf{R})$;
       set $\mathsf{R}=\mathsf{P}$ and $\mathsf{S}=\mathsf{dbl}(\mathsf{P})$;    else
       for $i=\ell-2,\ell-3,\ldots,0$ do       $\mathsf{R}=\mathsf{diffAdd}(\mathsf{R},\mathsf{S},\mathsf{P})$;
          $(\mathsf{R},\mathsf{S})=\mathsf{ladder}(\mathsf{R},\mathsf{S},n_i)$;       $\mathsf{S}=\mathsf{dbl}(\mathsf{S})$;
       return $(\mathsf{R},\mathsf{S})$.    return $(\mathsf{R},\mathsf{S})$.
     | Show Table
    DownLoad: CSV

    Table 3.  Some properties of the group of $\mathbb{F}_p$-rational points of the Legendre form elliptic curves $E_{1a}$, $E_{1b}$, $E_2$ and $E_3$

    $E_{1a}$ $E_{1b}$ $E_2$ $E_3$
    $p$ $2^{251}-9$ $2^{251}-9$ $2^{255}-19$ $2^{266}-3$
    $(\lg\ell,\lg\ell_T)$ $(248,248)$ $(248,248)$ $(251.4,252)$ $(262.4,263)$
    $(h,h_T)$ $(8,8)$ $(8,8)$ $(12,8)$ $(12,8)$
    $(k,k_T)$ $\left(\ell-1,\frac{\ell_T-1}{7}\right)$ $\left(\ell-1,\ell_T-1\right)$ $\left(\ell-1,\ell_T-1\right)$ $\left(\frac{\ell-1}{2},\ell_T-1\right)$
    $\lg (-D)$ $246.3$ $249.8$ $255$ $266$
    $\mathsf{KL}$ base pt $[64:1]$ $[19:1]$ $[31:1]$ $[2:1]$
     | Show Table
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    Table 4.  Conversions from Kummer line to Legendre form elliptic curves and vice versa. Here $\alpha_0 = \mathsf{a}^2$ and $\alpha_1 = \mathsf{b}^2$ are precomputed quantities

    KL to Legendre Legendre to KL
      $\widehat{\psi}([\mathsf{x}^2:\mathsf{z}^2])$ $\widehat{\psi}^{-1}(X:\cdot:Z)$
        $X=\alpha_0\mathsf{z}^2$;     $\mathsf{x}^2=\alpha_0(X-Z)$;
        $t_1=\alpha_1\mathsf{x}^2$;     $\mathsf{z}^2=\alpha_1X$;
        $Z=X-t_1$;   return $[\mathsf{x}^2:\mathsf{z}^2]$.
      return $(X:\cdot:Z)$.
     | Show Table
    DownLoad: CSV

    Table 5.  Base points for $E_{1a}$, $E_{1b}$, $E_2$ and $E_3$ corresponding to $\mathsf{KL}_{1a}$, $\mathsf{KL}_{1b}$, $\mathsf{KL}_{2}$ and $\mathsf{KL}_{3}$

    $p$ $\mathsf{a}^2$ $\mathsf{b}^2$ $[\mathsf{x}^2:\mathsf{z}^2]$ $(x,y)$
    $2^{251}-9$ $81$ $20$ $[64:1]$ $(-81/1199,\mathfrak{y}_1)$
    $2^{251}-9$ $186$ $175$ $[19:1]$ $(-186/3139,\mathfrak{y}_2)$
    $2^{255}-19$ $82$ $77$ $[31:1]$ $(-82/2305,\mathfrak{y}_3)$
    $2^{266}-3$ $260$ $139$ $[2:1]$ $(-260/18,\mathfrak{y}_4)$
     | Show Table
    DownLoad: CSV

    Table 6.  Values of $x_1,y_1$ and $x_2$ which are solutions to (24)

    $x_2=0$ $x_1 = \sqrt{\mu}$ $y_1 = \pm\sqrt{-\mu^2 + 2\mu^{3/2} - \mu}$
    $x_1 = -\sqrt{\mu}$ $y_1 = \pm\sqrt{-\mu^2 - 2\mu^{3/2} - \mu}$
    $x_2=1$ $x_1 = 1 + \sqrt{1-\mu}$ $y_1 = \pm (-1+\mu-\sqrt{1-\mu})$
    $x_1 = 1 - \sqrt{1-\mu}$ $y_1 = \pm (-1+\mu+\sqrt{1-\mu})$
    $x_2=\mu$ $x_1 = \mu + \sqrt{\mu^2-\mu}$ $y_1 = \pm \left(2\mu^3 + 2\mu^2\sqrt{\mu^2-\mu}-3\mu^2-2\mu\sqrt{\mu^2-\mu}+\mu\right)^{1/2}$
    $x_1 = \mu - \sqrt{\mu^2-\mu}$ $y_1 = \pm \left(2\mu^3 - 2\mu^2\sqrt{\mu^2-\mu}-3\mu^2+2\mu\sqrt{\mu^2-\mu}+\mu\right)^{1/2}$
     | Show Table
    DownLoad: CSV

    Table 7.  Summary of the different twisted Edwards form curve. Here b.r. denotes birational equivalence and 2-iso denotes 2-isogeny

    Kummer Legendre twisted Edwards Legendre to twisted Edwards
    $\mathsf{KL2519}(81,20)$ $E_{1a}$ $\mathsf{Ed}_{1a,1}$ b.r. (Thm 4.4)
    $\mathsf{Ed}_{1a,2}$ b.r. (Thm 4.4)
    $\mathsf{KL2519}(186,175)$ $E_{1b}$ $\mathsf{Ed}_{1b,1}$ b.r. (Thm 4.4)
    $\mathsf{Ed}_{1b,2}$ b.r. (Thm 4.4)
    $\mathsf{Ed}_{1b,3}$ 2-iso (Thm 4.5)
    $\mathsf{KL25519}(82,77)$ $E_{2}$ $\mathsf{Ed}_{2}$ 2-iso (Thm 4.5)
    $\mathsf{KL2663}(260,139)$ $E_{3}$ $\mathsf{Ed}_{3}$ 2-iso (Thm 4.5)
     | Show Table
    DownLoad: CSV

    Table 8.  General $d$

    $A \leftarrow (V_1 - U_1 ) \cdot (V_2 - U_2 )$,
    $B \leftarrow (V_1 + U_1 ) \cdot (V_2 + U_2 )$,
    $C \leftarrow (2d) T_1 \cdot T_2$,
    $D \leftarrow 2W_1 \cdot W_2$,
    $E \leftarrow B - A$,
    $F \leftarrow D - C$,
    $G \leftarrow D + C$,
    $H \leftarrow B + A$,
    $U_3 \leftarrow E \cdot F$,
    $V_3 \leftarrow G \cdot H$,
    $T_3 \leftarrow E \cdot H$,
    $W_3 \leftarrow F \cdot G$.
     | Show Table
    DownLoad: CSV

    Table 9.  $d = d_1/d_2$ with $d_1,d_2$ small

    $A \leftarrow (V_1 - U_1 ) \cdot (V_2 - U_2 )$,
    $B \leftarrow (V_1 + U_1 ) \cdot (V_2 + U_2 )$,
    $C \leftarrow (2d_1) T_1 \cdot T_2$,
    $D \leftarrow (2d_2) W_1 \cdot W_2$,
    $E \leftarrow d_2(B - A)$,
    $F \leftarrow D - C$,
    $G \leftarrow D + C$,
    $H \leftarrow d_2(B + A)$,
    $U_3 \leftarrow E \cdot F$,
    $V_3 \leftarrow G \cdot H$,
    $T_3 \leftarrow E \cdot H$,
    $W_3 \leftarrow F \cdot G$.
     | Show Table
    DownLoad: CSV
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