1 # GENERATED by tools/generators/rw.py from ../../../tools/generators/secp256k1_hand/scalarpow.src; edit the .src 2 import Base 3 import ../../../spec/lib/common.bend as C 4 import ../../../spec/crypto/secp256k1/field.bend as FS 5 import ../../../src/crypto/secp256k1/limbs.bend as L 6 import ../../../src/crypto/secp256k1/scalar.bend as S 7 import ../../lib/nat.bend as N 8 import ../../lib/logic.bend as Lg 9 import ../../lib/lemmas/proofs/nat_algebra.bend as NA 10 import ../../math/natural/arith.bend as NR 11 import ../../math/typed/width.bend as WW 12 import ./semiring.bend as SR 13 import ./limbs.bend as LV 14 import ./ineq.bend as I 15 import ./consts.bend as K 16 import ./bitsv.bend as BV 17 import ./modexp.bend as ME 18 import ./scalarops.bend as SO 19 import ./lits.bend as LT 20 21 # Exponentiation modulo n (src/crypto/secp256k1/scalar.bend pow, inv): 22 # square-and-multiply over the exponent's bits computes x^e mod n, and the 23 # bits of inv spell n - 2. 24 25 def m(+one: Nat) -> Nat: 26 1n+K.np(one) 27 28 # one square-and-multiply step 29 def step_e(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +hx: {FS.reduced(one, FS.order(one), x) == True{} : Bool}, +acc: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), acc) == True{} : Bool}, +e: Nat, +he: {LV.ev(acc) == Nat.mod(Nat.pow(LV.ev(x), e), 1n+K.np(one)) : Nat}, b: Nat, +hb: {Nat.is_lt(b, 2n) == True{} : Bool}) -> {LV.ev(S.pow_step(b, x, S.mul(acc, acc))) == Nat.mod(Nat.pow(LV.ev(x), Nat.add(Nat.mul(b, one), Nat.double(e))), 1n+K.np(one)) : Nat}: 30 match b: 31 case 0n: 32 %Equal.sym(Nat, LV.ev(S.mul(acc, acc)), Nat.mod(Nat.mul(LV.ev(acc), LV.ev(acc)), 1n+K.np(one)), SO.mul_e(one, h1, acc, acc, ha, ha)) : {_ == Nat.mod(Nat.pow(LV.ev(x), Nat.double(e)), 1n+K.np(one)) : Nat} 33 ME.sq_step(K.np(one), LV.ev(x), e, LV.ev(acc), he) 34 case 1n: 35 +sq = S.mul(acc, acc) 36 +es = Equal.trans(Nat, LV.ev(sq), Nat.mod(Nat.mul(LV.ev(acc), LV.ev(acc)), 1n+K.np(one)), Nat.mod(Nat.pow(LV.ev(x), Nat.double(e)), 1n+K.np(one)), SO.mul_e(one, h1, acc, acc, ha, ha), ME.sq_step(K.np(one), LV.ev(x), e, LV.ev(acc), he)) 37 %Equal.sym(Nat, Nat.mul(1n, one), one, SR.one_mul(one)) : {LV.ev(S.mul(S.mul(acc, acc), x)) == Nat.mod(Nat.pow(LV.ev(x), Nat.add(_, Nat.double(e))), 1n+K.np(one)) : Nat} 38 %Equal.sym(Nat, one, 1n, h1) : {LV.ev(S.mul(S.mul(acc, acc), x)) == Nat.mod(Nat.pow(LV.ev(x), Nat.add(_, Nat.double(e))), 1n+K.np(one)) : Nat} 39 %Equal.sym(Nat, Nat.add(1n, Nat.double(e)), Nat.add(Nat.double(e), 1n), NA.add_comm(1n, Nat.double(e))) : {LV.ev(S.mul(S.mul(acc, acc), x)) == Nat.mod(Nat.pow(LV.ev(x), _), 1n+K.np(one)) : Nat} 40 %Equal.sym(Nat, LV.ev(S.mul(S.mul(acc, acc), x)), Nat.mod(Nat.mul(LV.ev(S.mul(acc, acc)), LV.ev(x)), 1n+K.np(one)), SO.mul_e(one, h1, S.mul(acc, acc), x, SO.mul_r(one, h1, acc, acc, ha, ha), hx)) : {_ == Nat.mod(Nat.pow(LV.ev(x), Nat.add(Nat.double(e), 1n)), 1n+K.np(one)) : Nat} 41 ME.mul_step(K.np(one), LV.ev(x), Nat.double(e), LV.ev(S.mul(acc, acc)), es) 42 case 2n+ +k: 43 Empty.absurd({LV.ev(S.pow_step(2n+k, x, S.mul(acc, acc))) == Nat.mod(Nat.pow(LV.ev(x), Nat.add(Nat.mul(2n+k, one), Nat.double(e))), 1n+K.np(one)) : Nat}, N.lt_zero_absurd(k, hb)) 44 45 def step_r(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +hx: {FS.reduced(one, FS.order(one), x) == True{} : Bool}, +acc: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), acc) == True{} : Bool}, b: Nat, +hb: {Nat.is_lt(b, 2n) == True{} : Bool}) -> {FS.reduced(one, FS.order(one), S.pow_step(b, x, S.mul(acc, acc))) == True{} : Bool}: 46 match b: 47 case 0n: 48 SO.mul_r(one, h1, acc, acc, ha, ha) 49 case 1n: 50 SO.mul_r(one, h1, S.mul(acc, acc), x, SO.mul_r(one, h1, acc, acc, ha, ha), hx) 51 case 2n+ +k: 52 Empty.absurd({FS.reduced(one, FS.order(one), S.pow_step(2n+k, x, S.mul(acc, acc))) == True{} : Bool}, N.lt_zero_absurd(k, hb)) 53 54 def pow_r(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +hx: {FS.reduced(one, FS.order(one), x) == True{} : Bool}, bs: List<&2, Nat>, +acc: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), acc) == True{} : Bool}, +hb: {BV.all01(bs) == True{} : Bool}) -> {FS.reduced(one, FS.order(one), S.pow_go(x, bs, acc)) == True{} : Bool}: 55 match bs: 56 case Nil{}: 57 ha 58 case +b <> +t: 59 +h0 = Lg.and_left(Nat.is_lt(b, 2n), BV.all01(t), hb) 60 pow_r(one, h1, x, hx, t, S.pow_step(b, x, S.mul(acc, acc)), step_r(one, h1, x, hx, acc, ha, b, h0), Lg.and_right(Nat.is_lt(b, 2n), BV.all01(t), hb)) 61 62 # x^e for the exponent spelled by the bits, from acc = x^e0 63 def pow_e(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +hx: {FS.reduced(one, FS.order(one), x) == True{} : Bool}, bs: List<&2, Nat>, +acc: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), acc) == True{} : Bool}, +e: Nat, +he: {LV.ev(acc) == Nat.mod(Nat.pow(LV.ev(x), e), 1n+K.np(one)) : Nat}, +hb: {BV.all01(bs) == True{} : Bool}) -> {LV.ev(S.pow_go(x, bs, acc)) == Nat.mod(Nat.pow(LV.ev(x), BV.hv(bs, one, e)), 1n+K.np(one)) : Nat}: 64 match bs: 65 case Nil{}: 66 he 67 case +b <> +t: 68 +h0 = Lg.and_left(Nat.is_lt(b, 2n), BV.all01(t), hb) 69 pow_e(one, h1, x, hx, t, S.pow_step(b, x, S.mul(acc, acc)), step_r(one, h1, x, hx, acc, ha, b, h0), Nat.add(Nat.mul(b, one), Nat.double(e)), step_e(one, h1, x, hx, acc, ha, e, he, b, h0), Lg.and_right(Nat.is_lt(b, 2n), BV.all01(t), hb)) 70 71 # S.pow looks at x first 72 def pow_is(+one: Nat, x: List<&2, Nat>, +hx: {FS.reduced(one, FS.order(one), x) == True{} : Bool}, +bs: List<&2, Nat>, +acc: List<&2, Nat>) -> {S.pow(x, bs, acc) == S.pow_go(x, bs, acc) : List<&2, Nat>}: 73 match x: 74 case Nil{}: 75 Empty.absurd({S.pow([], bs, acc) == S.pow_go([], bs, acc) : List<&2, Nat>}, Lg.false_true(Lg.and_left(FS.limbs16(one, 16n, []), Nat.is_lt(FS.value(one, []), FS.order(one)), hx))) 76 case y <> t: 77 {==} 78 79 def pow_ev(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +hx: {FS.reduced(one, FS.order(one), x) == True{} : Bool}, +bs: List<&2, Nat>, +acc: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), acc) == True{} : Bool}, +e: Nat, +he: {LV.ev(acc) == Nat.mod(Nat.pow(LV.ev(x), e), 1n+K.np(one)) : Nat}, +hb: {BV.all01(bs) == True{} : Bool}) -> {LV.ev(S.pow(x, bs, acc)) == Nat.mod(Nat.pow(LV.ev(x), BV.hv(bs, one, e)), 1n+K.np(one)) : Nat}: 80 %Equal.sym(List<&2, Nat>, S.pow(x, bs, acc), S.pow_go(x, bs, acc), pow_is(one, x, hx, bs, acc)) : {LV.ev(_) == Nat.mod(Nat.pow(LV.ev(x), BV.hv(bs, one, e)), 1n+K.np(one)) : Nat} 81 pow_e(one, h1, x, hx, bs, acc, ha, e, he, hb) 82 83 def pow_rv(+one: Nat, +h1: {one == 1n : Nat}, +x: List<&2, Nat>, +hx: {FS.reduced(one, FS.order(one), x) == True{} : Bool}, +bs: List<&2, Nat>, +acc: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), acc) == True{} : Bool}, +hb: {BV.all01(bs) == True{} : Bool}) -> {FS.reduced(one, FS.order(one), S.pow(x, bs, acc)) == True{} : Bool}: 84 %Equal.sym(List<&2, Nat>, S.pow(x, bs, acc), S.pow_go(x, bs, acc), pow_is(one, x, hx, bs, acc)) : {FS.reduced(one, FS.order(one), _) == True{} : Bool} 85 pow_r(one, h1, x, hx, bs, acc, ha, hb) 86 87 # ---- one ---- 88 89 def one_lt(+one: Nat, +h1: {one == 1n : Nat}) -> {Nat.is_lt(1n, FS.order(one)) == True{} : Bool}: 90 SO.small_lt_n(one, h1, 1n, {==}) 91 92 def one_r(+one: Nat, +h1: {one == 1n : Nat}) -> {FS.reduced(one, FS.order(one), S.one()) == True{} : Bool}: 93 SO.small_r(one, h1, 1n, one_lt(one, h1)) 94 95 def one_e(+one: Nat, +h1: {one == 1n : Nat}, +x: Nat) -> {LV.ev(S.one()) == Nat.mod(Nat.pow(x, 0n), 1n+K.np(one)) : Nat}: 96 +ev = Equal.trans(Nat, LV.ev(S.one()), FS.value(one, S.one()), 1n, Equal.sym(Nat, FS.value(one, S.one()), LV.ev(S.one()), LV.ev_value(one, h1, S.one())), SO.small_val(one, h1, 1n, one_lt(one, h1))) 97 Equal.trans(Nat, LV.ev(S.one()), 1n, Nat.mod(Nat.pow(x, 0n), 1n+K.np(one)), ev, ME.pow0(K.np(one), x, I.lt_subst_r(1n, FS.order(one), 1n+K.np(one), K.order_eq(one, h1), one_lt(one, h1)))) 98 99 # ---- the exponent of inv: n - 2 ---- 100 101 def xn() -> List<&2, Nat>: 102 [48832n, 12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n] 103 104 def zeros7(+one: Nat) -> {FS.digits(one, [0n, 0n, 0n, 0n, 0n, 0n, 0n]) == 0n : Nat}: 105 %Equal.sym(Nat, Nat.mul(one, 0n), 0n, NA.mul_zero(one)) : {Nat.add(_, C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, 0n)))))))))))))) == 0n : Nat} 106 %Equal.sym(Nat, Nat.mul(one, 0n), 0n, NA.mul_zero(one)) : {Nat.add(0n, C.shift(16n, Nat.add(_, C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, 0n)))))))))))))) == 0n : Nat} 107 %Equal.sym(Nat, Nat.mul(one, 0n), 0n, NA.mul_zero(one)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(_, C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, 0n)))))))))))))) == 0n : Nat} 108 %Equal.sym(Nat, Nat.mul(one, 0n), 0n, NA.mul_zero(one)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(_, C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, 0n)))))))))))))) == 0n : Nat} 109 %Equal.sym(Nat, Nat.mul(one, 0n), 0n, NA.mul_zero(one)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(_, C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, 0n)))))))))))))) == 0n : Nat} 110 %Equal.sym(Nat, Nat.mul(one, 0n), 0n, NA.mul_zero(one)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(_, C.shift(16n, Nat.add(Nat.mul(one, 0n), C.shift(16n, 0n)))))))))))))) == 0n : Nat} 111 %Equal.sym(Nat, Nat.mul(one, 0n), 0n, NA.mul_zero(one)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(_, C.shift(16n, 0n)))))))))))))) == 0n : Nat} 112 %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, _))))))))))))) == 0n : Nat} 113 %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, _)))))))))))) == 0n : Nat} 114 %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, _))))))))))) == 0n : Nat} 115 %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, _)))))))))) == 0n : Nat} 116 %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, _))))))))) == 0n : Nat} 117 %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, _)))))))) == 0n : Nat} 118 %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, _))))))) == 0n : Nat} 119 %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, _)))))) == 0n : Nat} 120 %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, Nat.add(0n, _))))) == 0n : Nat} 121 %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {Nat.add(0n, C.shift(16n, Nat.add(0n, C.shift(16n, _)))) == 0n : Nat} 122 %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {Nat.add(0n, C.shift(16n, Nat.add(0n, _))) == 0n : Nat} 123 %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {Nat.add(0n, C.shift(16n, _)) == 0n : Nat} 124 %Equal.sym(Nat, C.shift(16n, 0n), 0n, WW.shift_zero(16n)) : {Nat.add(0n, _) == 0n : Nat} 125 %Equal.sym(Nat, Nat.add(0n, 0n), 0n, {==}) : {_ == 0n : Nat} 126 {==} 127 128 # the trailing zero limbs add nothing 129 def zeros_tail(+one: Nat) -> {FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]) == FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]) : Nat}: 130 %Equal.sym(Nat, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]), Nat.add(FS.digits(one, L.take(8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n])), LV.pos(8n, FS.digits(one, L.drop(8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n])))), K.digits_split(one, 8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n])) : {_ == FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]) : Nat} 131 %Equal.sym(Nat, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]), Nat.add(FS.digits(one, L.take(8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), LV.pos(8n, FS.digits(one, L.drop(8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])))), K.digits_split(one, 8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])) : {Nat.add(FS.digits(one, L.take(8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n])), LV.pos(8n, FS.digits(one, L.drop(8n, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n])))) == _ : Nat} 132 %Equal.sym(Nat, FS.digits(one, [0n, 0n, 0n, 0n, 0n, 0n, 0n]), 0n, zeros7(one)) : {Nat.add(FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]), LV.pos(8n, _)) == Nat.add(FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]), LV.pos(8n, 0n)) : Nat} 133 {==} 134 135 # c_n + 1 136 def xn_digits(+one: Nat, +h1: {one == 1n : Nat}) -> {FS.digits(one, xn()) == Nat.add(FS.cn(one), one) : Nat}: 137 %Equal.sym(Nat, Nat.mul(one, 48832n), Nat.add(one, Nat.mul(one, 48831n)), NA.mul_succ(one, 48831n)) : {Nat.add(_, C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]))) == Nat.add(Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), one) : Nat} 138 %Equal.sym(Nat, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n]), FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]), zeros_tail(one)) : {Nat.add(Nat.add(one, Nat.mul(one, 48831n)), C.shift(16n, _)) == Nat.add(Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), one) : Nat} 139 %Equal.sym(Nat, Nat.add(Nat.add(one, Nat.mul(one, 48831n)), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), Nat.add(one, Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])))), NA.add_assoc(one, Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])))) : {_ == Nat.add(Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), one) : Nat} 140 %Equal.sym(Nat, Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.mul(one, 48831n)), NA.add_comm(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])))) : {Nat.add(one, _) == Nat.add(Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), one) : Nat} 141 %Equal.sym(Nat, Nat.add(one, Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.mul(one, 48831n))), Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.add(one, Nat.mul(one, 48831n))), NA.add_swap(one, C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.mul(one, 48831n))) : {_ == Nat.add(Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), one) : Nat} 142 %Equal.sym(Nat, Nat.add(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n]))), Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.mul(one, 48831n)), NA.add_comm(Nat.mul(one, 48831n), C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])))) : {Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.add(one, Nat.mul(one, 48831n))) == Nat.add(_, one) : Nat} 143 %Equal.sym(Nat, Nat.add(Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.mul(one, 48831n)), one), Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.add(Nat.mul(one, 48831n), one)), NA.add_assoc(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.mul(one, 48831n), one)) : {Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.add(one, Nat.mul(one, 48831n))) == _ : Nat} 144 %Equal.sym(Nat, Nat.add(Nat.mul(one, 48831n), one), Nat.add(one, Nat.mul(one, 48831n)), NA.add_comm(Nat.mul(one, 48831n), one)) : {Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), Nat.add(one, Nat.mul(one, 48831n))) == Nat.add(C.shift(16n, FS.digits(one, [12233n, 41331n, 16429n, 24516n, 20663n, 8985n, 17745n, 1n])), _) : Nat} 145 {==} 146 147 def xn_limbs(+one: Nat, +h1: {one == 1n : Nat}) -> {FS.limbs16(one, 16n, xn()) == True{} : Bool}: 148 LT.lits16(one, h1, 16n, xn(), {==}) 149 150 # the bits of c_n + 1, most significant first 151 def xb() -> List<&2, Nat>: 152 [0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 153 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 154 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 155 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 1n, 156 0n, 1n, 0n, 0n, 0n, 1n, 0n, 1n, 0n, 1n, 0n, 1n, 0n, 0n, 0n, 1n, 0n, 0n, 1n, 0n, 0n, 0n, 1n, 1n, 0n, 0n, 0n, 1n, 1n, 0n, 0n, 1n, 157 0n, 1n, 0n, 1n, 0n, 0n, 0n, 0n, 1n, 0n, 1n, 1n, 0n, 1n, 1n, 1n, 0n, 1n, 0n, 1n, 1n, 1n, 1n, 1n, 1n, 1n, 0n, 0n, 0n, 1n, 0n, 0n, 158 0n, 1n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 0n, 1n, 0n, 1n, 1n, 0n, 1n, 1n, 0n, 1n, 0n, 0n, 0n, 0n, 1n, 0n, 1n, 1n, 1n, 0n, 0n, 1n, 1n, 159 0n, 0n, 1n, 0n, 1n, 1n, 1n, 1n, 1n, 1n, 0n, 0n, 1n, 0n, 0n, 1n, 1n, 0n, 1n, 1n, 1n, 1n, 1n, 0n, 1n, 1n, 0n, 0n, 0n, 0n, 0n, 0n] 160 161 def bits_xb() -> {L.bits(xn()) == xb() : List<&2, Nat>}: 162 {==} 163 164 def ib_lit() -> {S.inv_bits() == L.cbits(xb()) : List<&2, Nat>}: 165 {==} 166 167 def xb01() -> {BV.all01(xb()) == True{} : Bool}: 168 {==} 169 170 def xb_len() -> {List.length(&2, Nat, xb()) == 256n : Nat}: 171 {==} 172 173 # hv(xb) = c_n + 1 174 def xb_v(+one: Nat, +h1: {one == 1n : Nat}) -> {BV.hv(xb(), one, 0n) == Nat.add(FS.cn(one), one) : Nat}: 175 %Equal.sym(List<&2, Nat>, xb(), L.bits(xn()), Equal.sym(List<&2, Nat>, L.bits(xn()), xb(), bits_xb())) : {BV.hv(_, one, 0n) == Nat.add(FS.cn(one), one) : Nat} 176 %Equal.sym(Nat, BV.hv(L.bits(xn()), one, 0n), Nat.add(LV.pos(16n, 0n), FS.digits(one, xn())), BV.bits_v(one, h1, 16n, xn(), 0n, xn_limbs(one, h1))) : {_ == Nat.add(FS.cn(one), one) : Nat} 177 %Equal.sym(Nat, LV.pos(16n, 0n), 0n, LV.pos_zero(16n)) : {Nat.add(_, FS.digits(one, xn())) == Nat.add(FS.cn(one), one) : Nat} 178 %Equal.sym(Nat, FS.digits(one, xn()), Nat.add(FS.cn(one), one), xn_digits(one, h1)) : {Nat.add(0n, _) == Nat.add(FS.cn(one), one) : Nat} 179 %Equal.sym(Nat, Nat.add(0n, Nat.add(FS.cn(one), one)), Nat.add(FS.cn(one), one), SR.zero_add(Nat.add(FS.cn(one), one))) : {_ == Nat.add(FS.cn(one), one) : Nat} 180 {==} 181 182 def inv_h2(+one: Nat, +h1: {one == 1n : Nat}) -> {Nat.add(FS.cn(one), Nat.add(BV.hv(S.inv_bits(), one, 0n), 2n)) == Nat.add(FS.cn(one), 1n+K.np(one)) : Nat}: 183 +hcb = BV.cb(one, xb(), 0n, 0n, xb01()) 184 %Equal.sym(List<&2, Nat>, S.inv_bits(), L.cbits(xb()), ib_lit()) : {Nat.add(FS.cn(one), Nat.add(BV.hv(_, one, 0n), 2n)) == Nat.add(FS.cn(one), 1n+K.np(one)) : Nat} 185 %Equal.sym(Nat, Nat.add(FS.cn(one), 1n+K.np(one)), Nat.add(1n+K.np(one), FS.cn(one)), NA.add_comm(FS.cn(one), 1n+K.np(one))) : {Nat.add(FS.cn(one), Nat.add(BV.hv(L.cbits(xb()), one, 0n), 2n)) == _ : Nat} 186 %Equal.sym(Nat, Nat.add(1n+K.np(one), FS.cn(one)), C.shift(256n, one), K.hm_n(one, h1)) : {Nat.add(FS.cn(one), Nat.add(BV.hv(L.cbits(xb()), one, 0n), 2n)) == _ : Nat} 187 %Equal.sym(Nat, C.shift(256n, one), Nat.add(Nat.add(BV.hv(L.cbits(xb()), one, 0n), BV.hv(xb(), one, 0n)), one), Equal.sym(Nat, Nat.add(Nat.add(BV.hv(L.cbits(xb()), one, 0n), BV.hv(xb(), one, 0n)), one), C.shift(256n, one), hcb)) : {Nat.add(FS.cn(one), Nat.add(BV.hv(L.cbits(xb()), one, 0n), 2n)) == _ : Nat} 188 %Equal.sym(Nat, BV.hv(xb(), one, 0n), Nat.add(FS.cn(one), one), xb_v(one, h1)) : {Nat.add(FS.cn(one), Nat.add(BV.hv(L.cbits(xb()), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(xb()), one, 0n), _), one) : Nat} 189 %Equal.sym(Nat, one, 1n, h1) : {Nat.add(FS.cn(one), Nat.add(BV.hv(L.cbits(xb()), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), _)), one) : Nat} 190 %Equal.sym(Nat, one, 1n, h1) : {Nat.add(FS.cn(one), Nat.add(BV.hv(L.cbits(xb()), one, 0n), 2n)) == Nat.add(Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 1n)), _) : Nat} 191 %Equal.sym(Nat, Nat.add(FS.cn(one), Nat.add(BV.hv(L.cbits(xb()), one, 0n), 2n)), Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 2n)), NA.add_swap(FS.cn(one), BV.hv(L.cbits(xb()), one, 0n), 2n)) : {_ == Nat.add(Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 1n)), 1n) : Nat} 192 %Equal.sym(Nat, Nat.add(Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 1n)), 1n), Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(Nat.add(FS.cn(one), 1n), 1n)), NA.add_assoc(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 1n), 1n)) : {Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 2n)) == _ : Nat} 193 %Equal.sym(Nat, Nat.add(Nat.add(FS.cn(one), 1n), 1n), Nat.add(FS.cn(one), Nat.add(1n, 1n)), NA.add_assoc(FS.cn(one), 1n, 1n)) : {Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 2n)) == Nat.add(BV.hv(L.cbits(xb()), one, 0n), _) : Nat} 194 %Equal.sym(Nat, Nat.add(1n, 1n), 2n, {==}) : {Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), 2n)) == Nat.add(BV.hv(L.cbits(xb()), one, 0n), Nat.add(FS.cn(one), _)) : Nat} 195 {==} 196 197 def exp_inv(+one: Nat, +h1: {one == 1n : Nat}) -> {BV.hv(S.inv_bits(), one, 0n) == Nat.sub(1n+K.np(one), 2n) : Nat}: 198 +h = BV.hv(S.inv_bits(), one, 0n) 199 +e = NR.add_cancel(FS.cn(one), Nat.add(h, 2n), 1n+K.np(one), inv_h2(one, h1)) 200 %Equal.sym(Nat, 1n+K.np(one), Nat.add(BV.hv(S.inv_bits(), one, 0n), 2n), Equal.sym(Nat, Nat.add(BV.hv(S.inv_bits(), one, 0n), 2n), 1n+K.np(one), e)) : {BV.hv(S.inv_bits(), one, 0n) == Nat.sub(_, 2n) : Nat} 201 %Equal.sym(Nat, Nat.add(BV.hv(S.inv_bits(), one, 0n), 2n), Nat.add(2n, BV.hv(S.inv_bits(), one, 0n)), NA.add_comm(BV.hv(S.inv_bits(), one, 0n), 2n)) : {BV.hv(S.inv_bits(), one, 0n) == Nat.sub(_, 2n) : Nat} 202 %Equal.sym(Nat, Nat.sub(Nat.add(2n, BV.hv(S.inv_bits(), one, 0n)), 2n), BV.hv(S.inv_bits(), one, 0n), N.add_sub_cancel(2n, BV.hv(S.inv_bits(), one, 0n))) : {BV.hv(S.inv_bits(), one, 0n) == _ : Nat} 203 {==} 204 205 def inv_bits01(+one: Nat) -> {BV.all01(S.inv_bits()) == True{} : Bool}: 206 {==} 207 208 def inv_v(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), a) == True{} : Bool}) -> {FS.value(one, S.inv(a)) == FS.minv(FS.order(one), FS.value(one, a)) : Nat}: 209 %Equal.sym(Nat, FS.value(one, S.pow(a, S.inv_bits(), S.one())), LV.ev(S.pow(a, S.inv_bits(), S.one())), LV.ev_value(one, h1, S.pow(a, S.inv_bits(), S.one()))) : {_ == Nat.mod(Nat.pow(FS.value(one, a), Nat.sub(FS.order(one), 2n)), FS.order(one)) : Nat} 210 %Equal.sym(Nat, LV.ev(S.pow(a, S.inv_bits(), S.one())), Nat.mod(Nat.pow(LV.ev(a), BV.hv(S.inv_bits(), one, 0n)), 1n+K.np(one)), pow_ev(one, h1, a, ha, S.inv_bits(), S.one(), one_r(one, h1), 0n, one_e(one, h1, LV.ev(a)), inv_bits01(one))) : {_ == Nat.mod(Nat.pow(FS.value(one, a), Nat.sub(FS.order(one), 2n)), FS.order(one)) : Nat} 211 %Equal.sym(Nat, BV.hv(S.inv_bits(), one, 0n), Nat.sub(1n+K.np(one), 2n), exp_inv(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), _), 1n+K.np(one)) == Nat.mod(Nat.pow(FS.value(one, a), Nat.sub(FS.order(one), 2n)), FS.order(one)) : Nat} 212 %Equal.sym(Nat, FS.value(one, a), LV.ev(a), LV.ev_value(one, h1, a)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.sub(1n+K.np(one), 2n)), 1n+K.np(one)) == Nat.mod(Nat.pow(_, Nat.sub(FS.order(one), 2n)), FS.order(one)) : Nat} 213 %Equal.sym(Nat, FS.order(one), 1n+K.np(one), K.order_eq(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.sub(1n+K.np(one), 2n)), 1n+K.np(one)) == Nat.mod(Nat.pow(LV.ev(a), Nat.sub(_, 2n)), FS.order(one)) : Nat} 214 %Equal.sym(Nat, FS.order(one), 1n+K.np(one), K.order_eq(one, h1)) : {Nat.mod(Nat.pow(LV.ev(a), Nat.sub(1n+K.np(one), 2n)), 1n+K.np(one)) == Nat.mod(Nat.pow(LV.ev(a), Nat.sub(1n+K.np(one), 2n)), _) : Nat} 215 {==} 216 217 def inv_r(+one: Nat, +h1: {one == 1n : Nat}, +a: List<&2, Nat>, +ha: {FS.reduced(one, FS.order(one), a) == True{} : Bool}) -> {FS.reduced(one, FS.order(one), S.inv(a)) == True{} : Bool}: 218 pow_rv(one, h1, a, ha, S.inv_bits(), S.one(), one_r(one, h1), inv_bits01(one))