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This commit is contained in:
Dmitry Vagner 2023-02-07 15:09:49 -08:00
parent 69afed9297
commit cb7c638cbc
2 changed files with 53 additions and 40 deletions
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74
evm/src/cpu/kernel/asm/curve/bn254/curve_arithmetic/invariant_exponent.asm
View File

@ -1,37 +1,17 @@
/// To make the Tate pairing an invariant, the final step is to exponentiate by
/// (p^12 - 1)/N = (p^6 - 1)(p^2 + 1)(p^4 - p^2 + 1)/N
/// The function in this module enacts the final exponentiation, by
/// (p^4 - p^2 + 1)/N = p^3 + (a2)p^2 - (a1)p - a0
/// (p^12 - 1)/N = (p^6 - 1) * (p^2 + 1) * (p^4 - p^2 + 1)/N
/// and thus we can exponentiate by each factor sequentially.
///
/// def final_exp(y):
/// y4, y2, y0 = 1, 1, 1
/// power_loop_4()
/// power_loop_2()
/// power_loop_0()
/// custom_powers()
/// final_power()
///
/// def custom_powers()
/// y0 = y0^{-1}
/// y1 = y4 * y2^2 * y0
/// return y2, y1, y0
///
/// def final_power()
/// y = y.frob(3)
/// y2 = y2.frob(2)
/// y1 = y1.frob(1)
/// return y * y2 * y1 * y0
/// def bn254_invariant_exponent(y: Fp12):
/// y = first_exp(y)
/// y = second_exp(y)
/// return final_exponentiation(y)
/// return final_exp(y)
global bn254_invariant_exponent:
/// map t to t^(p^6 - 1) via
/// def first_exp(t):
/// return t.frob(6) / t
/// first, exponentiate by (p^6 - 1) via
/// def first_exp(y):
/// return y.frob(6) / y
// stack: out, retdest {out: y}
%stack (out) -> (out, 100, first_exp, out)
// stack: out, 100, first_exp, out, retdest {out: y}
@ -44,9 +24,9 @@ first_exp:
// stack: out, 100, out, second_exp, out, retdest {out: y_6, 100: y^-1}
%jump(mul_fp254_12)
/// map t to t^(p^2 + 1) via
/// def second_exp(t):
/// return t.frob(2) * t
/// second, exponentiate by (p^2 + 1) via
/// def second_exp(y):
/// return y.frob(2) * y
second_exp:
// stack: out, retdest {out: y}
%stack (out) -> (out, 100, out, out, final_exp, out)
@ -55,6 +35,22 @@ second_exp:
// stack: 100, out, out, final_exp, out, retdest {out: y, 100: y_2}
%jump(mul_fp254_12)
/// Finally, we must exponentiate by (p^4 - p^2 + 1)/N
/// To do so efficiently, we can express this power as
/// (p^4 - p^2 + 1)/N = p^3 + (a2)p^2 - (a1)p - a0
/// and simultaneously compute y^a4, y^a2, y^a0 where
/// a1 = a4 + 2a2 - a0
/// We first initialize these powers as 1 and then use
/// binary algorithms for exponentiation.
///
/// def final_exp(y):
/// y4, y2, y0 = 1, 1, 1
/// power_loop_4()
/// power_loop_2()
/// power_loop_0()
/// custom_powers()
/// final_power()
final_exp:
// stack: val, retdest
%stack (val) -> (val, 300, val)
@ -71,6 +67,25 @@ final_exp:
// stack: 64, 62, 65, 300, val, retdest {200: y0, 212: y2, 224: y4}
%jump(power_loop_4)
/// After computing the powers
/// y^a4, y^a2, y^a0
/// we would like to transform them to
/// y^a2, y^-a1, y^-a0
///
/// def custom_powers()
/// y0 = y0^{-1}
/// y1 = y4 * y2^2 * y0
/// return y2, y1, y0
///
/// And finally, upon doing so, compute the final power
/// y^(p^3) * (y^a2)^(p^2) * (y^-a1)^p * (y^-a0)
///
/// def final_power()
/// y = y.frob(3)
/// y2 = y2.frob(2)
/// y1 = y1.frob(1)
/// return y * y2 * y1 * y0
custom_powers:
// stack: val, retdest {200: y0, 212: y2, 224: y4}
%stack () -> (200, 236, make_term_1)
@ -91,7 +106,6 @@ make_term_3:
%stack () -> (236, 224, 224, final_power)
// stack: 236, 224, 224, final_power, val, retdest {212: y2, 224: y4 * y2^2, 236: y0^-1}
%jump(mul_fp254_12)
final_power:
// stack: val, retdest {val: y , 212: y^a2 , 224: y^a1 , 236: y^a0}
%frob_fp254_12_3

19
evm/src/cpu/kernel/asm/curve/bn254/curve_arithmetic/tate_pairing.asm
View File

@ -1,3 +1,12 @@
/// def tate(P: Curve, Q: TwistedCurve) -> Fp12:
/// out = miller_loop(P, Q)
/// return bn254_invariant_exponent(P, Q)
global bn254_tate:
// stack: inp, out, retdest
%stack (inp, out) -> (inp, out, bn254_invariant_exponent, out)
// stack: inp, out, bn254_invariant_exponent, out, retdest
%jump(bn254_miller)
/// def miller(P, Q):
/// miller_init()
/// miller_loop()
@ -24,16 +33,6 @@
/// def miller_zero():
/// 0xnm -= 1
/// mul_tangent()
/// def tate(P: Curve, Q: TwistedCurve) -> Fp12:
/// out = miller_loop(P, Q)
/// return bn254_invariant_exponent(P, Q)
global bn254_tate:
// stack: inp, out, retdest
%stack (inp, out) -> (inp, out, bn254_invariant_exponent, out)
// stack: inp, out, bn254_invariant_exponent, out, retdest
%jump(bn254_miller)
global bn254_miller:
// stack: ptr, out, retdest
%stack (ptr, out) -> (out, 1, ptr, out)