use.std::math::ecgfp5::base_field
#! Given an encoded elliptic curve point `w` s.t. it's expressed using
#! an element ∈ GF(p^5) | p = 2^64 - 2^32 + 1, this routine verifies whether
#! given point can be successfully decoded or not
#!
#! Expected stack state
#!
#! [w0, w1, w2, w3, w4, ...]
#!
#! Final stack state
#!
#! [flg, ...]
#!
#! If w can be decoded, flg = 1
#! Else flg = 0
#!
#! Note, if w = (0, 0, 0, 0, 0), it can be successfully decoded to point
#! at infinity i.e. flg = 1, in that case.
#!
#! See https://github.com/pornin/ecgfp5/blob/ce059c6/python/ecGFp5.py#L1043-L1052
#! for reference implementation
export.validate
repeat.5
dup.4
end
exec.base_field::square
sub.2 # = e
exec.base_field::square
swap
sub.1052
swap # = delta
exec.base_field::legendre
eq.1
movdn.5
push.1
repeat.5
swap
eq.0
and
end
or
end
#! Given an encoded elliptic curve point `w` s.t. it's expressed using
#! an element ∈ GF(p^5) | p = 2^64 - 2^32 + 1, this routine attempts to decode
#! it into x, y coordinates, along with boolean field element denoting whether it's
#! point-at-infinity or not.
#!
#! Expected stack state
#!
#! [w0, w1, w2, w3, w4, ...]
#!
#! Final state state
#!
#! [x0, x1, x2, x3, x4, y0, y1, y2, y3, y4, inf, flg, ...]
#!
#! If `w` has be decoded, flg = 1
#! Else flg = 0 and x, y = (0, 0)
#!
#! Note, when w = (0, 0, 0, 0, 0), it will be successfully decoded to
#! point-at-infinity i.e. x, y = (0, 0) and flg = 1
#!
#! See https://github.com/pornin/ecgfp5/blob/ce059c6/python/ecGFp5.py#L1022-L1041
#! for reference implementation
export.decode
repeat.5
dup.4
end
exec.base_field::square
sub.2 # = e
repeat.5
dup.4
end
exec.base_field::square
swap
sub.1052
swap # = delta
exec.base_field::sqrt # = (r, c)
repeat.5
dup.10
end
repeat.5
dup.9
end
exec.base_field::add
push.0.0.0.0.9223372034707292161
exec.base_field::mul # = x1
repeat.5
movup.9
end
repeat.5
movup.15
end
exec.base_field::sub
push.0.0.0.0.9223372034707292161
exec.base_field::mul # = x2
repeat.5
movup.9
end
repeat.5
dup.4
end
exec.base_field::legendre
eq.1
if.true
repeat.5
movup.5
drop
end
else
repeat.5
drop
end
end # = x
repeat.5
dup.10
end
repeat.5
dup.9
end
exec.base_field::mul
repeat.5
movup.4
neg
end # = y
dup.10
not # = inf
push.1
repeat.5
movup.13
eq.0
and
end
movup.12
or # = c
swap
repeat.5
movup.6
end
repeat.5
movup.11
end
add.6148914689804861441 # = x
end
#! Given an elliptic curve point as Weierstraß coordinates (X, Y) along with
#! boolean field element `inf`, denoting whether this is point-at-infinity or not,
#! this routine encodes it to a single element ∈ GF(p^5) | p = 2^64 - 2^32 + 1
#!
#! Expected stack state
#!
#! [x0, x1, x2, x3, x4, y0, y1, y2, y3, y4, inf, ...]
#!
#! Final stack state
#!
#! [w0, w1, w2, w3, w4, ...]
#!
#! Note, when inf = 1, encoded point w = (0, 0, 0, 0, 0)
#!
#! See https://github.com/pornin/ecgfp5/blob/ce059c6/python/ecGFp5.py#L1214-L1216
#! for reference implementation.
export.encode
push.0.0.0.0.6148914689804861441 # = a/ 3
exec.base_field::sub # = (a/ 3) - x
repeat.5
movup.9
end
exec.base_field::div # = w = y/ ((a/ 3) - x)
movup.5
if.true
repeat.5
drop
end
push.0.0.0.0.0
end
end
#! Given two elliptic curve points ( say a, b ) as Weierstraß coordinates (X, Y) on stack,
#! this routine computes elliptic curve point c, resulting from a + b.
#!
#! Following point addition formula is complete and it works when two points are
#! same/ different or input operands are point-at-infinity.
#!
#! Expected stack state
#!
#! [x1_0, x1_1, x1_2, x1_3, x1_4, y1_0, y1_1, y1_2, y1_3, y1_4, inf1, x2_0, x2_1, x2_2, x2_3, x2_4, y2_0, y2_1, y2_2, y2_3, y2_4, inf2, ...]
#!
#! s.t. x1_{0..5} -> x1, y1_{0..5} -> y1 |> a = (x1, y1, inf1)
#! x2_{0..5} -> x2, y2_{0..5} -> y2 |> b = (x2, y2, inf2)
#!
#! Final stack state
#!
#! [x3_0, x3_1, x3_2, x3_3, x3_4, y3_0, y3_1, y3_2, y3_3, y3_4, inf3, ...]
#!
#! Read point addition section ( on page 8 ) of https://ia.cr/2022/274
#! For reference implementation see https://github.com/pornin/ecgfp5/blob/ce059c6/python/ecGFp5.py#L1228-L1255
export.add.40
loc_storew_be.0
dropw
loc_store.4 # cached x1
loc_storew_be.8
dropw
loc_store.12 # cached y1
loc_store.16 # cached inf1
loc_storew_be.20
dropw
loc_store.24 # cached x2
loc_storew_be.28
dropw
loc_store.32 # cached y2
loc_store.36 # cached inf2
loc_load.24
push.0.0.0.0
loc_loadw_be.20 # bring x2
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x1
exec.base_field::eq
dup
if.true
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x1
exec.base_field::square
repeat.5
movup.4
mul.3
end
add.6148914689804861439
swap
add.263
swap
else
loc_load.12
push.0.0.0.0
loc_loadw_be.8 # bring y1
loc_load.32
push.0.0.0.0
loc_loadw_be.28 # bring y2
exec.base_field::sub
end # = λ0
dup.5
if.true
loc_load.12
push.0.0.0.0
loc_loadw_be.8 # bring y1
repeat.5
movup.4
mul.2
end
else
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x1
loc_load.24
push.0.0.0.0
loc_loadw_be.20 # bring x2
exec.base_field::sub
end # = λ1
repeat.5
movup.9
end
exec.base_field::div # = λ
repeat.5
dup.4
end
exec.base_field::square # = λ^2
loc_load.24
push.0.0.0.0
loc_loadw_be.20 # bring x2
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x1
exec.base_field::add
repeat.5
movup.9
end
exec.base_field::sub # compute x3
repeat.5
dup.4
end
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x1
exec.base_field::sub
repeat.5
movup.14
end
exec.base_field::mul
loc_load.12
push.0.0.0.0
loc_loadw_be.8 # bring y1
repeat.5
movup.9
end
exec.base_field::sub # compute y3
movup.10
loc_load.12
push.0.0.0.0
loc_loadw_be.8 # bring y1
loc_load.32
push.0.0.0.0
loc_loadw_be.28 # bring y2
exec.base_field::neq
and # compute inf3
movdn.5
# finalize selection of y3
loc_load.32
push.0.0.0.0
loc_loadw_be.28 # bring y2
loc_load.16 # bring inf1
if.true
repeat.5
movup.5
drop
end
else
repeat.5
drop
end
end
loc_load.12
push.0.0.0.0
loc_loadw_be.8 # bring y1
loc_load.36 # bring inf2
if.true
repeat.5
movup.5
drop
end
else
repeat.5
drop
end
end
# finalize selection of x3
repeat.5
movup.10
end
loc_load.24
push.0.0.0.0
loc_loadw_be.20 # bring x2
loc_load.16 # bring inf1
if.true
repeat.5
movup.5
drop
end
else
repeat.5
drop
end
end
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x1
loc_load.36 # bring inf2
if.true
repeat.5
movup.5
drop
end
else
repeat.5
drop
end
end
# finalize selection of inf3
movup.10
loc_load.36 # bring inf2
loc_load.16 # bring inf1
cdrop
loc_load.16 # bring inf1
loc_load.36 # bring inf2
cdrop
movdn.10
end
#! Given one elliptic curve point ( say a ) as Weierstraß coordinates (X, Y) on stack,
#! this routine computes elliptic curve point b s.t. b = 2 * a.
#!
#! Following point doubling formula is complete and it works only when input operand is
#! a non-infinity point, then resulting point b should also be non-infinity.
#!
#! Note, result of add(a, b) = double(a) | a = b
#!
#! Expected stack state
#!
#! [x0, x1, x2, x3, x4, y0, y1, y2, y3, y4, inf, ...]
#!
#! s.t. x{0..5} -> x, y{0..5} -> y |> a = (x, y, inf)
#!
#! Final stack state
#!
#! [x'0, x'1, x'2, x'3, x'4, y'0, y'1, y'2, y'3, y'4, inf, ...]
#!
#! Read point addition section ( on page 8 ) of https://ia.cr/2022/274
#! For reference implementation see https://github.com/pornin/ecgfp5/blob/ce059c6/python/ecGFp5.py#L1270-L1280
export.double.20
loc_storew_be.0
dropw
loc_store.4 # cached x
loc_storew_be.8
dropw
loc_store.12 # cached y
loc_store.16 # cached inf
loc_load.12
push.0.0.0.0
loc_loadw_be.8 # bring y
repeat.5
movup.4
mul.2
end # compute λ1
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x
exec.base_field::square
repeat.5
movup.4
mul.3
end
add.6148914689804861439
swap
add.263
swap # compute λ0
exec.base_field::div # compute λ
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x
repeat.5
movup.4
mul.2
end
repeat.5
dup.9
end
exec.base_field::square
exec.base_field::sub # compute x'
repeat.5
dup.4
end
loc_load.4
push.0.0.0.0
loc_loadw_be.0 # bring x
exec.base_field::sub
repeat.5
movup.14
end
exec.base_field::mul
loc_load.12
push.0.0.0.0
loc_loadw_be.8 # bring y
repeat.5
movup.9
end
exec.base_field::sub # compute y'
repeat.5
movup.9
end
loc_load.16
movdn.10
end
#! Given an elliptic curve point ( say a ) as Weierstraß coordinates (X, Y) and a 319 -bit scalar ( say e )
#! on stack, this routine computes elliptic curve point b s.t. b = e * a, using double-and-add technique.
#!
#! Scalar e should be lesser than 1067993516717146951041484916571792702745057740581727230159139685185762082554198619328292418486241 ( prime number ).
#! Note, scalar e should be provided as 10 limbs on stack, each of 32 -bit, representing it in radix-2^32 form.
#!
#! Given a scalar e ( as arbitrary width big integer ), following python code snippet should convert it to desired input form
#!
#! [(a >> (32*i)) & 0xffff_ffff for i in range(10)]
#!
#! Expected stack state
#!
#! [x0, x1, x2, x3, x4, y0, y1, y2, y3, y4, inf, e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, ...]
#!
#! Point a = (x, y, inf)
#! Scalar e = (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9)
#!
#! Final stack state
#!
#! [x'0, x'1, x'2, x'3, x'4, y'0, y'1, y'2, y'3, y'4, inf, ...]
#!
#! Point b = (x', y' inf') | b = e * a
#!
#! See https://github.com/itzmeanjan/secp256k1/blob/cbbe199/point.py#L174-L186 for source of inpiration.
export.mul.40
loc_storew_be.0
dropw
loc_store.4 # cached base_x
loc_storew_be.8
dropw
loc_store.12 # cached base_y
loc_store.16 # cached base_inf
push.0.0.0.0
loc_storew_be.20
dropw
push.0
loc_store.24 # initialize and cache res_x
push.0.0.0.0
loc_storew_be.28
dropw
push.0
loc_store.32 # initialize and cache res_y
push.1
loc_store.36 # initialize and cache res_inf
repeat.10
repeat.32
dup
push.1
u32and
if.true
# bring base
loc_load.16
loc_load.12
push.0.0.0.0
loc_loadw_be.8
loc_load.4
push.0.0.0.0
loc_loadw_be.0
# bring res
loc_load.36
loc_load.32
push.0.0.0.0
loc_loadw_be.28
loc_load.24
push.0.0.0.0
loc_loadw_be.20
exec.add
# write back res
loc_storew_be.20
dropw
loc_store.24
loc_storew_be.28
dropw
loc_store.32
loc_store.36
end
# bring base
loc_load.16
loc_load.12
push.0.0.0.0
loc_loadw_be.8
loc_load.4
push.0.0.0.0
loc_loadw_be.0
exec.double
# write back base
loc_storew_be.0
dropw
loc_store.4
loc_storew_be.8
dropw
loc_store.12
loc_store.16
u32shr.1
end
drop
end
# bring res
loc_load.36
loc_load.32
push.0.0.0.0
loc_loadw_be.28
loc_load.24
push.0.0.0.0
loc_loadw_be.20
end
#! Given a 319 -bit scalar ( say e ) on stack, this routine computes elliptic curve point
#! b s.t. b = e * G, using double-and-add technique | G = conventional group generator point.
#!
#! Group generator point https://github.com/pornin/ecgfp5/blob/ce059c6/rust/src/curve.rs#L67-L83
#!
#! Scalar e should be lesser than N ( = 1067993516717146951041484916571792702745057740581727230159139685185762082554198619328292418486241 ).
#! Note, scalar e should be provided as 10 limbs on stack, each of 32 -bit, representing it in radix-2^32 form.
#!
#! Given a 319 -bit scalar e, following python code snippet should convert it to
#! desired input form i.e. radix-2^32 representation, having ten 32 -bit limbs
#!
#! [(e >> (32*i)) & 0xffff_ffff for i in range(10)]
#!
#! Expected stack state
#!
#! [e0, e1, e2, e3, e4, e5, e6, e7, e8, e9, ...]
#!
#! Scalar e = (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9)
#!
#! Final stack state
#!
#! [x0, x1, x2, x3, x4, y0, y1, y2, y3, y4, inf, ...]
#!
#! Point b = (x, y, inf) | b = e * G
#!
#! See https://github.com/itzmeanjan/secp256k1/blob/cbbe199/point.py#L174-L186 for source of inpiration.
export.gen_mul.32
loc_storew_be.0
dropw
loc_storew_be.4
dropw
push.0.0
movup.3
movup.3
loc_storew_be.8
dropw # cache 319 -bit scalar, having 10 limbs, each of 32 -bit width
push.0.0.0.0.0
loc_storew_be.12
dropw
loc_store.16 # initialize and cache res_x
push.0.0.0.0.0
loc_storew_be.20
dropw
loc_store.24 # initialize and cache res_y
push.1
loc_store.28 # initialize and cache res_inf
push.0.8623373087021137817.12280941316336150722.16498106075800537106.17561351883262171669.14191612778916076371.18286383079848041790.5863339128417000744.11451759428629996034.14179124258460441178.4768624678334228939
push.0.16248535820734672314.12843058077554438509.14604603115652880441.1861370543582612103.12413761366396265525.4176022593458990041.3883454194479152564.14208568149880154517.30496475895846659.10372352552633189742
push.0.2877809984845595236.8124729848406989261.2903444924486455483.6139401771336820486.11258486910517561409.11269995078679563195.4075713964805364983.11272694349658053421.17286345916541825175.5239921069304273244
push.0.17641185070824107975.14889445785061292430.7405791732550181507.4195033912468829473.5893024420900179836.9198199668142261635.13255077950461830861.3315714462260585145.5358622364758307393.721593612819002908
push.0.17218153792253988554.398898134320123335.14691321837712781231.14813642725547346961.7022468048599869463.13425501193148363047.8102520424076163562.4818391030611537206.3117791970810252983.3664867151911665097
push.0.9768740144997694838.18337438909916942052.13287690569608682189.1749912136742008160.16874873892224416289.13753848043314362441.4990704782108972448.15940429530085642075.10014152261700797213.8271724616203496328
push.0.15137575290038364568.834448034475692680.17725871744982540755.17720761966581073520.15617784849739721933.13143846705867213452.6387719880838102748.15637132133232092065.17395668299430845447.4976508673457147220
push.0.14448869278120871938.2440192432902894712.9065627619858283302.7476463190805008531.12433592364249876790.7326369368968030334.10492708378186270656.8590652592090351714.11478014260122655233.5102808632757846047
push.0.9652768576975789451.2169826252445647515.8515692953826676326.12790755226033428483.10499532484372206399.1752808319082265622.1461287152438866664.16156917747677348345.1913114203733512691.4220966228724218121
push.0.15678769274395291067.3644569546435216629.8279510540289007754.14347910311032351181.11017801090920149683.17092277456226211965.16027019793415245260.6242090196981371796.13195765456550520156.18329307820651300136
push.0.15660277884414351657.4747038350935563407.16563445468461400225.9202649824211012050.16989226289451019470.16269357368883802812.14601257932273902054.11922650544629087011.3837671014910122842.10542030917794106031
push.0.16679682878508344885.4260113432582040236.6657572838446037989.17057619578187970416.9557888942099085229.16766876175803959905.8797349954628290450.9735674650584728206.16707168790898826743.2957163603751591175
push.0.12731950592067067806.15132000389362928180.2301212109204980010.11858193693385253107.2787225233368024015.15608358023222707950.5664305148663915403.14236763768002019539.11718445275434263967.10887359501593613247
push.0.12906063996958603900.1561861388017124445.9769071188656310541.9108206805299149795.828335823705130147.11630764285144089035.13798633568618668074.10120521355198906717.78547975849802565.9032713792450238388
push.0.2932017870248423748.12862241115656865401.5691212362156829751.10513473720672864882.5811042078412067511.458337204753456641.13679640590869584883.3442439947769535104.10679288079271307010.16403385062095364376
push.0.2960040779968781381.13743098123458581162.16796911018466257282.3963340258774352059.9493732769165536890.17565729304514950270.14558776703731825254.16603544858443598146.1290832334148046531.7196562104405968541
push.0.12121575497725661883.16189154896554038370.713702076670502983.2511512527285855394.13074864091280271682.11226573117410114168.4334454122606443001.7823766160466615569.18258517404340224673.12260059464719152256
push.0.6346101384116280016.17164907193019931643.6403805942282737631.2083047126212086181.15943828273518998540.14240418992504638979.959753680948073425.8797147096983019241.7835785575038815398.15427624465125434692
push.0.289487472969449713.7875160791236525843.12277145790039167649.18340877018641812993.6669569099586313850.14033385862453802.1174914055793702087.17415867199848812284.12669785880287947992.17495080701629382991
push.0.8312474118415594556.17907937811574560916.10199001911441156238.11406737362222088800.2978515999920985364.14117192075707323389.10658282210448969656.16696421864516574633.622946243598315577.6442977565493598461
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repeat.2 # process first 8 limbs of scalar
repeat.4 # process 4 consecutive 32 -bit limbs
repeat.32 # process each 32 -bit limb
push.0.0.0.0
loc_loadw_be.0
dup
push.1
u32and
movdn.4
u32shr.1
loc_storew_be.0
dropw
if.true
# base already on stack top
# bring res from memory
loc_load.28
loc_load.24
push.0.0.0.0
loc_loadw_be.20
loc_load.16
push.0.0.0.0
loc_loadw_be.12
exec.add
# write back res
loc_storew_be.12
dropw
loc_store.16
loc_storew_be.20
dropw
loc_store.24
loc_store.28
else
dropw
dropw
push.0
dropw
end
end
push.0.0.0.0
loc_loadw_be.0
drop
push.0
movdn.3
loc_storew_be.0
dropw
end
push.0.0.0.0
loc_loadw_be.4
loc_storew_be.0
dropw
end
repeat.2 # process last 2 limbs of scalar
repeat.32 # process each of last two 32 -bit limbs
push.0.0.0.0
loc_loadw_be.8
dup
push.1
u32and
movdn.4
u32shr.1
loc_storew_be.8
dropw
if.true
# base already on stack top
# bring res from memory
loc_load.28
loc_load.24
push.0.0.0.0
loc_loadw_be.20
loc_load.16
push.0.0.0.0
loc_loadw_be.12
exec.add
# write back res
loc_storew_be.12
dropw
loc_store.16
loc_storew_be.20
dropw
loc_store.24
loc_store.28
else
dropw
dropw
push.0
dropw
end
end
push.0.0.0.0
loc_loadw_be.8
drop
push.0
movdn.3
loc_storew_be.8
dropw
end
# bring res back to stack
loc_load.28
loc_load.24
push.0.0.0.0
loc_loadw_be.20
loc_load.16
push.0.0.0.0
loc_loadw_be.12
end