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123 lines
2.3 KiB
123 lines
2.3 KiB
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/*
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Source: https://en.wikipedia.org/wiki/Montgomery_curve
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1 + y 1 + y
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[u, v] = [ ------- , ---------- ]
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1 - y (1 - y)x
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*/
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template Edwards2Montgomery() {
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signal input in[2];
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signal output out[2];
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out[0] <-- (1 + in[1]) / (1 - in[1]);
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out[1] <-- out[0] / in[0];
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out[0] * (1-in[1]) === (1 + in[1]);
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out[1] * in[0] === out[0];
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}
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/*
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u u - 1
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[x, y] = [ ---, ------- ]
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v u + 1
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*/
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template Montgomery2Edwards() {
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signal input in[2];
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signal output out[2];
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out[0] <-- in[0] / in[1];
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out[1] <-- (in[0] - 1) / (in[0] + 1);
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out[0] * in[1] === in[0];
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out[1] * (in[0] + 1) === in[0] - 1;
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}
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/*
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x2 - x1
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lamda = ---------
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y2 - y1
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x3 + A + x1 + x2
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x3 = B * lamda^2 - A - x1 -x2 => lamda^2 = ------------------
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B
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y3 = (2*x1 + x2 + A)*lamda - B*lamda^3 - y1 =>
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=> y3 = lamda * ( 2*x1 + x2 + A - x3 - A - x1 - x2) - y1 =>
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=> y3 = lamda * ( x1 - x3 ) - y1
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----------
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y2 - y1
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lamda = ---------
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x2 - x1
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x3 = B * lamda^2 - A - x1 -x2
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y3 = lamda * ( x1 - x3 ) - y1
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*/
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template MontgomeryAdd() {
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signal input in1[2];
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signal input in2[2];
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signal output out[2];
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var a = 168700;
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var d = 168696;
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var A = (2 * (a + d)) / (a - d);
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var B = 4 / (a - d);
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signal lamda;
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lamda <-- (in2[1] - in1[1]) / (in2[0] - in1[0]);
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lamda * (in2[0] - in1[0]) === (in2[1] - in1[1]);
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out[0] <== B*lamda*lamda - A - in1[0] -in2[0];
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out[1] <== lamda * (in1[0] - out[0]) - in1[1];
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}
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/*
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x1_2 = x1*x1
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3*x1_2 + 2*A*x1 + 1
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lamda = ---------------------
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2*B*y1
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x3 = B * lamda^2 - A - x1 -x1
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y3 = lamda * ( x1 - x3 ) - y1
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*/
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template MontgomeryDouble() {
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signal input in[2];
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signal output out[2];
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var a = 168700;
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var d = 168696;
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var A = (2 * (a + d)) / (a - d);
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var B = 4 / (a - d);
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signal lamda;
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signal x1_2;
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x1_2 <== in[0] * in[0];
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lamda <-- (3*x1_2 + 2*A*in[0] + 1 ) / (2*B*in[1]);
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lamda * (2*B*in[1]) === (3*x1_2 + 2*A*in[0] + 1 );
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out[0] <== B*lamda*lamda - A - 2*in[0];
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out[1] <== lamda * (in[0] - out[0]) - in[1];
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}
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