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scalar_p.sx
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scalar_p.sx
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.section .rodata
.global prime_p_and_c
prime_p_and_c: .quad 0xfffffffefffffc2f, 0x1000003d1
.text
.include "aliases"
.include "commonmacros"
.include "u256macros"
/**
Tests whether a scalar in sa0...sa3 is zero.
Returns 0 in io0 if true, otherwise 1.
*/
BEGIN_GLOBAL_FUNCTION a_is_zero
IS_A_ZERO
END_GLOBAL_FUNCTION a_is_zero
/**
Tests whether a scalar is zero.
Input:
Address of scalar in io0
Result:
io0: 0 if scalar is zero, otherwise 1
*/
BEGIN_C_FUNCTION secp256k1_is_zero
LOAD_A
IS_A_ZERO
END_C_FUNCTION secp256k1_is_zero
/**
Tests whether a Scalar r (lo, sr0-sr3) is zero.
Returns 0 in io0 if true, otherwise 1.
*/
BEGIN_GLOBAL_FUNCTION r_is_zero
IS_R_LO_ZERO
END_GLOBAL_FUNCTION r_is_zero
/**
The macro REDUCE_512_MOD_P
reduces a 512-bit-value r to a 256-bit-value mod p.
p = 2**256 - c, with c = ppc = 0x1000003d1
==> 2**256 = p+c = 0+c =c
*/
.macro REDUCE_512_MOD_P
//sr7*ppc to sr4 sr3
reduce_r7\@:
mul tmp4, ppc, sr7 //c*r7 lo
umulh tmp5, ppc, sr7 //c*r7 hi
adds sr3, sr3, tmp4
adcs sr4, sr4, tmp5
adcs sr5, sr5, xzr
adcs sr6, sr6, xzr
adc sr7, xzr, xzr //new carry in sr7?
cbnz sr7, reduce_r7\@
//sr6*ppc to sr3 sr2
reduce_r6\@:
mul tmp4, ppc, sr6 //c*r6 lo
umulh tmp5, ppc, sr6 //c*r6 hi
adds sr2, sr2, tmp4
adcs sr3, sr3, tmp5
adcs sr4, sr4, xzr
adcs sr5, sr5, xzr
adc sr6, xzr, xzr //new carry in sr6?
cbnz sr6, reduce_r6\@
//sr5*ppc to sr2 sr1
reduce_r5\@: //r5=sr5
mul tmp4, ppc, sr5 //c*r5 lo
umulh tmp5, ppc, sr5 //c*r5 hi
adds sr1, sr1, tmp4
adcs sr2, sr2, tmp5
adcs sr3, sr3, xzr
adcs sr4, sr4, xzr
adc sr5, xzr, xzr //new carry in sr5?
cbnz sr5, reduce_r5\@
//sr4*ppc to sr1 sr0
reduce_r4\@: //r4=sr4
mul tmp4, ppc, sr4 //c*r4 lo
umulh tmp5, ppc, sr4 //c*r4 hi
adds sr0, sr0, tmp4
adcs sr1, sr1, tmp5
adcs sr2, sr2, xzr
adcs sr3, sr3, xzr
adc sr4, xzr, xzr //new carry in sr4?
cbnz sr4, reduce_r4\@
REDUCE_256_MOD_P
.endm
/*
The macro REDUCE_256_MOD_P
reduces a 256-bit-value x, if x>= p.
Asigns the reduced value to x.
x is expected in sr0...sr3
Output:
in sr0...sr3
*/
.macro REDUCE_256_MOD_P reg0=sr0 reg1=sr1 reg2=sr2 reg3=sr3
cmp \reg3, pp1
blo is_lower\@
bhi is_higher\@
cmp \reg2, pp1
blo is_lower\@
bhi is_higher\@
cmp \reg1, pp1
blo is_lower\@
bhi is_higher\@
cmp \reg0, pp0
blo is_lower\@
//here x >= n
is_higher\@: //x > prime -> subtract prime
subs \reg0, \reg0, pp0
sbcs \reg1, \reg1, pp1
sbcs \reg2, \reg2, pp1
sbc \reg3, \reg3, pp1
is_lower\@: //x < prime, do nothing
.endm
/*
Macro ADD_A_B_MOD_P
Modular Addition modulo p
r = (a + b) mod p
a in sa0...sa3
b in sb0...sb3
r in sr0...sr3
The prime is expected in pp0 and pp1
*/
.macro ADD_A_B_MOD_P
adds sr0, sa0, sb0
adcs sr1, sa1, sb1
adcs sr2, sa2, sb2
adcs sr3, sa3, sb3
adc sr4, xzr, xzr
cbz sr4, possibly_reduce\@
carry_occured\@:
subs sr0, sr0, pp0
sbcs sr1, sr1, pp1
sbcs sr2, sr2, pp1
sbcs sr3, sr3, pp1
adc sr4, xzr, xzr
cbnz sr4, carry_occured\@ //this is possible if a > prime and b > prime
possibly_reduce\@: //reduces if sum >= prime
REDUCE_256_MOD_P
.endm
/** Wrapper function for internal use*/
BEGIN_GLOBAL_FUNCTION add_a_b_mod_p
ADD_A_B_MOD_P
END_GLOBAL_FUNCTION add_a_b_mod_p
/*
The public C compatible function secp256k1_add_mod_p adds two scalars a + b
modulo p.
r = (a + b) mod p
Address of scalar a in io0
Address of scalar b in io1
Address of result r in io2
*/
BEGIN_C_FUNCTION secp256k1_add_mod_p
INIT_PRIME_P
LOAD_A_B
ADD_A_B_MOD_P
STORE_R_LO io2
END_C_FUNCTION secp256k1_add_mod_p
/*
Macro ADD_A_B_MOD_P_ASIGN_A
Modular Addition modulo p
r = (a + b) mod p
a = r
a in sa0...sa3
b in sb0...sb3
r in sr0...sr3
The prime is expected in pp0 and pp1
*/
.macro ADD_A_B_MOD_P_ASIGN_A
ADD_A_B_MOD_P
COPY_R_LO_TO_A
.endm
/** Wrapper function */
BEGIN_GLOBAL_FUNCTION add_a_b_mod_p_asign_a
ADD_A_B_MOD_P_ASIGN_A
END_GLOBAL_FUNCTION add_a_b_mod_p_asign_a
/*
Macro ADD_A_B_MOD_P_ASIGN_B
Modular Addition modulo p
r = (a + b) mod p
b = r
a in sa0...sa3
b in sb0...sb3
r in sb0...sb3
The prime is expected in pp0 and pp1
*/
.macro ADD_A_B_MOD_P_ASIGN_B
ADD_A_B_MOD_P
COPY_R_LO_TO_B
.endm
/** Wrapper function */
BEGIN_GLOBAL_FUNCTION add_a_b_mod_p_asign_b
ADD_A_B_MOD_P_ASIGN_B
END_GLOBAL_FUNCTION add_a_b_mod_p_asign_b
/*
Macro DOUBLE_A_MOD_P
Modular duplication modulo p
r = (a + a) mod p
a in sa0...sa3
r in sr0...sr3
The prime is expected in pp0 and pp1
*/
.macro DOUBLE_A_MOD_P
adds sr0, sa0, sa0
adcs sr1, sa1, sa1
adcs sr2, sa2, sa2
adcs sr3, sa3, sa3
adc sr4, xzr, xzr
cbz sr4, possibly_reduce_\@
carry_occured_\@:
subs sr0, sr0, pp0
sbcs sr1, sr1, pp1
sbcs sr2, sr2, pp1
sbcs sr3, sr3, pp1
adc sr4, xzr, xzr
cbnz sr4, carry_occured_\@ //this is possible if a > prime and b > prime
possibly_reduce_\@: //reduces if sum >= prime
REDUCE_256_MOD_P
.endm
/** Wrapper function */
BEGIN_GLOBAL_FUNCTION double_a_mod_p
DOUBLE_A_MOD_P
END_GLOBAL_FUNCTION double_a_mod_p
/** Public C function */
BEGIN_C_FUNCTION secp256k1_double_mod_p
INIT_PRIME_P
LOAD_A
DOUBLE_A_MOD_P
STORE_R_LO io1
END_C_FUNCTION secp256k1_double_mod_p
/** Public C function */
BEGIN_C_FUNCTION secp256k1_double_mod_p_asign
INIT_PRIME_P
LOAD_A
DOUBLE_A_MOD_P
STORE_R_LO io0
END_C_FUNCTION secp256k1_double_mod_p_asign
/*
Macro DOUBLE_A_MOD_P_ASIGN_A
Modular duplication modulo p
r = (a + a) mod p
a = r
a in sa0...sa3
r in sb0...sb3
The prime is expected in pp0 and pp1
*/
.macro DOUBLE_A_MOD_P_ASIGN_A
DOUBLE_A_MOD_P
COPY_R_LO_TO_A
.endm
/** Wrapper function */
BEGIN_GLOBAL_FUNCTION double_a_mod_p_asign_a
DOUBLE_A_MOD_P_ASIGN_A
END_GLOBAL_FUNCTION double_a_mod_p_asign_a
/*
Macro DOUBLE_A_MOD_P_ASIGN_B
Modular duplication modulo p
r = (a + a) mod p
b = r
a in sa0...sa3
b in sb0...sb3
r in sb0...sb3
The prime is expected in pp0 and pp1
*/
.macro DOUBLE_A_MOD_P_ASIGN_B
DOUBLE_A_MOD_P
COPY_R_LO_TO_B
.endm
/** Wrapper function */
BEGIN_GLOBAL_FUNCTION double_a_mod_p_asign_b
DOUBLE_A_MOD_P_ASIGN_B
END_GLOBAL_FUNCTION double_a_mod_p_asign_b
/*
Macro TRIPLE_A_MOD_P
Modular triplication modulo p
r = (a + a + a) mod p
a in sa0...sa3
r in sr0...sr3
The prime is expected in pp0 and pp2
*/
.macro TRIPLE_A_MOD_P
adds sr0, sa0, sa0
adcs sr1, sa1, sa1
adcs sr2, sa2, sa2
adcs sr3, sa3, sa3
adc sr4, xzr, xzr
adds sr0, sr0, sa0
adcs sr1, sr1, sa1
adcs sr2, sr2, sa2
adcs sr3, sr3, sa3
adc sr4, sr4, xzr
cbz sr4, _possibly_reduce_\@
_carry_occured_\@:
mul sr5, sr4, ppc //carries*c
adds sr0, sr0, sr5
adcs sr1, sr1, xzr
adcs sr2, sr2, xzr
adcs sr3, sr3, xzr
adc sr4, xzr, xzr
cbnz sr4, _carry_occured_\@ //this is possible if a > prime and b > prime
_possibly_reduce_\@: //reduces if sum >= prime
REDUCE_256_MOD_P
.endm
/** wrapper function*/
BEGIN_GLOBAL_FUNCTION triple_a_mod_p
TRIPLE_A_MOD_P
END_GLOBAL_FUNCTION triple_a_mod_p
/** public C function */
BEGIN_C_FUNCTION secp256k1_triple_mod_p
INIT_PRIME_P
LOAD_A
TRIPLE_A_MOD_P
STORE_R_LO io1
END_C_FUNCTION secp256k1_triple_mod_p
/** public C function */
BEGIN_C_FUNCTION secp256k1_triple_mod_p_asign
INIT_PRIME_P
LOAD_A
TRIPLE_A_MOD_P
STORE_R_LO io0
END_C_FUNCTION secp256k1_triple_mod_p_asign
/*
Macro TRIPLE_A_MOD_P_ASIGN_A
Modular triplication modulo p
r = (a + a + a) mod p
a = r
a in sa0...sa3
r in sr0...sr3
The prime is expected in pp0 and pp1
*/
.macro TRIPLE_A_MOD_P_ASIGN_A
TRIPLE_A_MOD_P
COPY_R_LO_TO_A
.endm
/** wrapper function*/
BEGIN_GLOBAL_FUNCTION triple_a_mod_p_asign_a
TRIPLE_A_MOD_P_ASIGN_A
END_GLOBAL_FUNCTION triple_a_mod_p_asign_a
/*
Macro TRIPLE_A_MOD_P_ASIGN_B
Modular triplication modulo p
r = (a + a + a) mod p
b = r
a in sa0...sa3
b in sb0...sb3
r in sr0...sr3
The prime is expected in pp0 and pp1
*/
.macro TRIPLE_A_MOD_P_ASIGN_B
TRIPLE_A_MOD_P
COPY_R_LO_TO_B
.endm
/** wrapper function*/
BEGIN_GLOBAL_FUNCTION triple_a_mod_p_asign_b
TRIPLE_A_MOD_P_ASIGN_B
END_GLOBAL_FUNCTION triple_a_mod_p_asign_b
//**********************subtraction modulo p********************************
/*
Macro SUB_A_B_MOD_P
Modular subtraction modulo p
r = (a - b) mod n
a in sa0...sa3
b in sb0...sb3
r in sr0...sr3
The prime is expected in pp0 and pp1
*/
.macro SUB_A_B_MOD_P
subs sr0, sa0, sb0
sbcs sr1, sa1, sb1
sbcs sr2, sa2, sb2
sbcs sr3, sa3, sb3
sbc sr4, xzr, xzr
cbz sr4, possibly_reduce\@
//carry occured -> add p and ignore carry
adds sr0, sr0, pp0
adcs sr1, sr1, pp1
adcs sr2, sr2, pp1
adc sr3, sr3, pp1
b endsubmod\@
possibly_reduce\@:
REDUCE_256_MOD_P
endsubmod\@:
.endm
/**wrapper functions*/
BEGIN_GLOBAL_FUNCTION sub_a_b_mod_p
SUB_A_B_MOD_P
END_GLOBAL_FUNCTION sub_a_b_mod_p
BEGIN_GLOBAL_FUNCTION sub_a_b_mod_p_asign_a
SUB_A_B_MOD_P
mov sa0, sr0
mov sa1, sr1
mov sa2, sr2
mov sa3, sr3
END_GLOBAL_FUNCTION sub_a_b_mod_p_asign_a
BEGIN_GLOBAL_FUNCTION sub_a_b_mod_p_asign_b
SUB_A_B_MOD_P
mov sb0, sr0
mov sb1, sr1
mov sb2, sr2
mov sb3, sr3
END_GLOBAL_FUNCTION sub_a_b_mod_p_asign_b
/** Public C function */
BEGIN_C_FUNCTION secp256k1_sub_mod_p
INIT_PRIME_P
LOAD_A_B
SUB_A_B_MOD_P
STORE_R_LO io2
END_C_FUNCTION secp256k1_sub_mod_p
/**
a = -a mod p
*/
.macro negate_a_mod_p
subs sr0, xzr, sa0
sbcs sr1, xzr, sa1
sbcs sr2, xzr, sa2
sbcs sr3, xzr, sa3
sbc sr4, xzr, xzr
cbz sr4, possibly_reduce\@
//carry occured -> add p and ignore carry
adds sr0, sr0, pp0
adcs sr1, sr1, pp1
adcs sr2, sr2, pp1
adc sr3, sr3, pp1
b endsubmod\@
possibly_reduce\@:
REDUCE_256_MOD_P
endsubmod\@:
.endm
/** Public C function */
BEGIN_C_FUNCTION secp256k1_negate_mod_p
INIT_PRIME_P
LOAD_A
REDUCE_256_MOD_P sa0 sa1 sa2 sa3
negate_a_mod_p
STORE_R_LO io1
END_C_FUNCTION secp256k1_negate_mod_p
BEGIN_C_FUNCTION secp256k1_negate_mod_p_asign
INIT_PRIME_P
LOAD_A
REDUCE_256_MOD_P sa0 sa1 sa2 sa3
negate_a_mod_p
STORE_R_LO io0
END_C_FUNCTION secp256k1_negate_mod_p_asign
/** Internal functions */
BEGIN_GLOBAL_FUNCTION square_a_mod_p
SQUARE_U256
REDUCE_512_MOD_P
END_GLOBAL_FUNCTION square_a_mod_p
BEGIN_GLOBAL_FUNCTION square_a_mod_p_asign_a
SQUARE_U256
REDUCE_512_MOD_P
COPY_R_LO_TO_A
END_GLOBAL_FUNCTION square_a_mod_p_asign_a
BEGIN_GLOBAL_FUNCTION square_a_mod_p_asign_b
SQUARE_U256
REDUCE_512_MOD_P
COPY_R_LO_TO_B
END_GLOBAL_FUNCTION square_a_mod_p_asign_b
/** Public C functions */
BEGIN_C_FUNCTION secp256k1_square_mod_p
INIT_PRIME_P
LOAD_A
SQUARE_U256
REDUCE_512_MOD_P
STORE_R_LO io1
END_C_FUNCTION secp256k1_square_mod_p
BEGIN_C_FUNCTION secp256k1_square_mod_p_asign
INIT_PRIME_P
LOAD_A
SQUARE_U256
REDUCE_512_MOD_P
STORE_R_LO io0
END_C_FUNCTION secp256k1_square_mod_p_asign
/**
Macro MUL_A_B_MOD_P
multiplies two Scalars modulo p
r= (a*b) mod p
Input:
a: in sa0-sa3
b: in sb0-sb3
Output:
r: in sr0-sr3
*/
.macro MUL_A_B_MOD_P
MUL_U256
REDUCE_512_MOD_P
.endm
/** wrapper functions */
BEGIN_GLOBAL_FUNCTION mul_a_b_mod_p
MUL_A_B_MOD_P
END_GLOBAL_FUNCTION mul_a_b_mod_p
BEGIN_GLOBAL_FUNCTION mul_a_b_mod_p_asign_b
MUL_A_B_MOD_P
COPY_R_LO_TO_B
END_GLOBAL_FUNCTION mul_a_b_mod_p_asign_b
BEGIN_GLOBAL_FUNCTION mul_a_b_mod_p_asign_a
MUL_A_B_MOD_P
COPY_R_LO_TO_A
END_GLOBAL_FUNCTION mul_a_b_mod_p_asign_a
/**
Public C function secp256k1_mul_mod_p
r = (a*b) mod p
Input:
Address of a in io0
Address of b in io1
Address of r in io2
*/
BEGIN_C_FUNCTION secp256k1_mul_mod_p
INIT_PRIME_P
LOAD_A_B
MUL_A_B_MOD_P
STORE_R_LO io2
END_C_FUNCTION secp256k1_mul_mod_p
/**
Macro CUBE_A_MOD_P cubes scalar a
r = a³
Input: a in sa0...sa3
Result: r in sr0...sr3
*/
.macro CUBE_A_MOD_P
mov sb0, sa0
mov sb1, sa1
mov sb2,sa2
mov sb3,sa3
bl square_a_mod_p_asign_a
bl mul_a_b_mod_p
.endm
/** wrapper function */
BEGIN_GLOBAL_FUNCTION cube_a_mod_p
CUBE_A_MOD_P
END_GLOBAL_FUNCTION cube_a_mod_p
BEGIN_GLOBAL_FUNCTION cube_a_mod_p_asign_a
CUBE_A_MOD_P
COPY_R_LO_TO_A
END_GLOBAL_FUNCTION cube_a_mod_p_asign_a
BEGIN_GLOBAL_FUNCTION cube_a_mod_p_asign_b
CUBE_A_MOD_P
COPY_R_LO_TO_B
END_GLOBAL_FUNCTION cube_a_mod_p_asign_a
/** public C function */
BEGIN_C_FUNCTION secp256k1_cube_mod_p
INIT_PRIME_P
LOAD_A io0
CUBE_A_MOD_P
STORE_R_LO io1
END_C_FUNCTION secp256k1_cube_mod_p
BEGIN_C_FUNCTION secp256k1_cube_mod_p_asign
INIT_PRIME_P
LOAD_A io0
CUBE_A_MOD_P
STORE_R_LO io0
END_C_FUNCTION secp256k1_cube_mod_p_asign
/*
Macros for the functions that use the same exponentiation algorithm but
with different exponents.
*/
.macro BIT0 //(exp & 1 != 1)
bl square_a_mod_p_asign_a
.endm
.macro BIT1 //(exp & 1 == 1)
bl mul_a_b_mod_p_asign_b
bl square_a_mod_p_asign_a
.endm
.macro NIB0 //0x0
BIT0
BIT0
BIT0
BIT0
.endm
.macro NIB1 //0x1
BIT1
BIT0
BIT0
BIT0
.endm
.macro NIB2 //0x2
BIT0
BIT1
BIT0
BIT0
.endm
.macro NIB3 //0x3
BIT1
BIT1
BIT0
BIT0
.endm
.macro NIB7 //0x7
BIT1
BIT1
BIT1
BIT0
.endm
.macro NIBb //0xb
BIT1
BIT1
BIT0
BIT1
.endm
.macro NIBc //0xc
BIT0
BIT0
BIT1
BIT1
.endm
.macro NIBd //0xd
BIT1
BIT0
BIT1
BIT1
.endm
.macro NIBe //0xe
BIT0
BIT1
BIT1
BIT1
.endm
.macro NIBf //0xf
BIT1
BIT1
BIT1
BIT1
.endm
.macro BYTEff //ff
NIBf
NIBf
.endm
.macro QUADX //fffffffefffffc2d
NIBd
NIB2
NIBc
NIBf
BYTEff
BYTEff
NIBe
NIBf
BYTEff
BYTEff
BYTEff
.endm
.macro QUADY //0xFFFFFFFFBFFFFF0C
NIBc
NIB0
BYTEff
BYTEff
NIBf
NIBb
BYTEff
BYTEff
BYTEff
BYTEff
.endm
.macro QUADZ //ffffffff7ffffe17
NIB7
NIB1
NIBe
BYTEff
BYTEff
NIB7
BYTEff
BYTEff
BYTEff
BYTEff
.endm
.macro QUADff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
.endm
.macro QUAD3f
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
NIBf
NIB3
.endm
.macro QUAD7f
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
BYTEff
NIBf
NIB7
.endm
/*
Macro INVERT_A_MOD_P_INTO_B
calculates the inverse of scalar a modulo p
b = inv a = a^-1 = a^p-2 (mod p)
--> a*b = a*(inv a) = 1
Input
a: sa0 sa1 sa2 sa3
Result
b: sb0 sb1 sb2 sb3
Algorithm (all functions are modulo p):
exp=0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2d //p-2
b = 1 //The result
for(int i = 0; i<256; i++){
if (exp & 1 == 1){
b = b * a mod p;
}
a = a * a mod p;
exp = exp >> 1;
}
In INVERT_A_MOD_P_INTO_B the for-loop, anding and shifting of the exponent (p-2) are
unrolled using the macros above.
*/
.macro INVERT_A_MOD_P_INTO_B
SET_B_TO_1
QUADX
QUADff
QUADff
QUADff
.endm
/** wrapper function */
BEGIN_GLOBAL_FUNCTION invert_a_mod_p_into_b
INVERT_A_MOD_P_INTO_B
END_GLOBAL_FUNCTION invert_a_mod_p_into_b
/** C function */
BEGIN_C_FUNCTION secp256k1_invert_mod_p
INIT_PRIME_P
LOAD_A
INVERT_A_MOD_P_INTO_B
STORE_B io1
END_C_FUNCTION secp256k1_invert_mod_p
/** C function */
BEGIN_C_FUNCTION secp256k1_invert_mod_p_asign
INIT_PRIME_P
LOAD_A
INVERT_A_MOD_P_INTO_B
STORE_B io0
END_C_FUNCTION secp256k1_invert_mod_p_asign
/*
Function exponentiate_a_e_mod_p
calculates the base a raised to the power of exponent e modulo prime p
b = a^e (mod p)
Input
a: sa0...sa3
e: k0...k3
Result
b: sb0 sb1 sb2 sb3
Algorithm (all functions are modulo p):
b = 1 //The result
for(int i = 0; i<256; i++){ //LOOP_EXPONENT
if (e & 1 == 1){
b = a * b mod p; //function mul_A_B_mod_p_asign_b
}
a = a * a mod p; //function square_a_mod_p_asign_a
e = e >> 1;
}
*/
.macro LOOP_EXPONENT register
mov io0, #64
loop_exp_\@:
cbz io0, endloop_exp_\@
and io3, \register, #1
//if (exp & 1) == 1 -> b = a*b mod p
cbz io3, biszero_\@
bl mul_a_b_mod_p_asign_b
biszero_\@:
bl square_a_mod_p_asign_a
lsr \register, \register, #1
sub io0, io0, #1
b loop_exp_\@
endloop_exp_\@:
.endm
BEGIN_GLOBAL_FUNCTION exponentiate_a_e_mod_p
SET_B_TO_1
LOOP_EXPONENT sk0
LOOP_EXPONENT sk1
LOOP_EXPONENT sk2
LOOP_EXPONENT sk3
END_GLOBAL_FUNCTION exponentiate_a_e_mod_p
BEGIN_C_FUNCTION secp256k1_exponentiate_mod_p
INIT_PRIME_P
LOAD_A
LOAD_K io1
bl exponentiate_a_e_mod_p
STORE_B io2
END_C_FUNCTION secp256k1_exponentiate_mod_p
BEGIN_GLOBAL_FUNCTION squareroot_a_mod_p_asign_b
SET_B_TO_1
QUADY
QUADff
QUADff
QUAD3f
END_GLOBAL_FUNCTION squareroot_a_mod_p_asign_b
/*
Function secp256k1_squareroot_mod_p
calculates the square root b of a scalar a modulo p.
e = (p+1)/4 = (p+1)*inv 4 = 0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFBFFFFF0C
b = a^e = squareroot of a
a = b² mod p
The result is undefined if a has no squareroot.
Input
a: sa0 sa1 sa2 sa3
Result
b: sb0 sb0 sb2 sb3
Algorithm (all functions are modulo p):
exp=0x3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFBFFFFF0C //(p+1)/4
b = 1 //The result
for(int i = 0; i<256; i++){
if (exp & 1 == 1){
b = b * a mod p; //function secp256k1_mul_ab_asign_b
}
a = a * a mod p; //function secp256k1_square_asign_a
exp = exp >> 1;
In secp256k1_squareroot_private the for-loop, anding and shifting of the
exponent are unrolled using the macros above.
}
*/
BEGIN_C_FUNCTION secp256k1_squareroot_mod_p
INIT_PRIME_P
LOAD_A
bl squareroot_a_mod_p_asign_b
STORE_B io1
END_C_FUNCTION secp256k1_squareroot_mod_p
BEGIN_C_FUNCTION secp256k1_squareroot_mod_p_asign
INIT_PRIME_P
LOAD_A
bl squareroot_a_mod_p_asign_b
STORE_B io0
END_C_FUNCTION secp256k1_squareroot_mod_p_asign
/**
Function secp256k1_has_squareroot_mod_p
calculates Eulers Criterion b = a^(p-1)/2 = a^((p-1)>>1)
The b is 1 if a has a squareroot otherwise p-1
Algorithm (all functions are modulo p):
exp=0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffff7ffffe17 //(p-1)/2
b = 1
for(int i = 0; i<256; i++){
if (exp & 1 == 1){
b = b * a mod p;
}
a = a * a mod p;
exp = exp >> 1;
In has_square_root_private the for-loop, anding and shifting of the
exponent are unrolled using the macros above.
}
*/
BEGIN_C_FUNCTION secp256k1_has_squareroot_mod_p
INIT_PRIME_P
LOAD_A
SET_B_TO_1
QUADZ
QUADff
QUADff
QUAD7f
mov io0, sb0
END_C_FUNCTION secp256k1_has_squareroot_mod_p