Data update

2026-04-30 12:34:36 -04:00 · 2026-04-30 12:34:36 -04:00 · cbaf4c4b64
commit cbaf4c4b64
parent 4bb20c9b71
12390 changed files with 318560 additions and 27248 deletions
--- a/Task/Tonelli-Shanks-algorithm/Erlang/tonelli-shanks-algorithm.erl
+++ b/Task/Tonelli-Shanks-algorithm/Erlang/tonelli-shanks-algorithm.erl
@ -0,0 +1,113 @@
+-module(tonelli_shanks).
+-export([tonelli/2, legendre/2, task/1,  main/1]).
+
+%% Public API
+tonelli(N, P) ->
+    case legendre(N, P) of
+        1 ->
+            {Q, S} = decompose(P - 1, 0),
+            if
+                S =:= 1 ->
+                    mod_pow(N, (P + 1) div 4, P);
+                S > 1 ->
+                    Z = find_non_square(P),
+                    C = mod_pow(Z, Q, P),
+                    R = mod_pow(N, (Q + 1) div 2, P),
+                    T = mod_pow(N, Q, P),
+                    tonelli_loop(P, C, R, T, S)
+            end;
+        _ ->
+            error({not_a_square_mod_p, N, P})
+    end.
+
+%% Tonelli-Shanks loop
+tonelli_loop(P, _C, R, T, _M) when T =:= 1 ->
+    R;
+tonelli_loop(P, C, R, T, M) ->
+    {T2, I} = find_least_i(T, 1, M, P),
+    B = mod_pow(C, mod_pow(2, M - I - 1, P - 1), P),
+    C1 = mod_mul(B, B, P),
+    R1 = mod_mul(R, B, P),
+    T1 = mod_mul(T, C1, P),
+    tonelli_loop(P, C1, R1, T1, I).
+
+%% Find smallest i such that t^(2^i) ≡ 1 mod p
+find_least_i(T, I, M, P) when I < M ->
+    T2 = mod_mul(T, T, P),
+    if
+        T2 =:= 1 -> {T2, I};
+        true -> find_least_i(T2, I + 1, M, P)
+    end;
+find_least_i(_, _, _, _) ->
+    error(no_i_found).
+
+%% Legendre symbol: a^((p-1)/2) mod p
+legendre(A, P) ->
+    mod_pow(A, (P - 1) div 2, P).
+
+%% Decompose p-1 = q * 2^s where q is odd
+decompose(Q, S) when Q rem 2 =:= 0 ->
+    decompose(Q div 2, S + 1);
+decompose(Q, S) ->
+    {Q, S}.
+
+%% Find a quadratic non-residue (Legendre = p - 1)
+find_non_square(P) ->
+    find_non_square(2, P).
+
+find_non_square(Z, P) ->
+    case legendre(Z, P) of
+        L when L =:= P - 1 -> Z;
+        _ -> find_non_square(Z + 1, P)
+    end.
+
+%% Modular exponentiation: base^exp mod mod
+mod_pow(Base, Exp, Mod) ->
+    mod_pow(Base, Exp, Mod, 1).
+
+mod_pow(_Base, 0, _Mod, Acc) ->
+    Acc;
+mod_pow(Base, Exp, Mod, Acc) ->
+    Acc1 = case Exp rem 2 of
+               1 -> (Acc * Base) rem Mod;
+               0 -> Acc
+           end,
+    Base2 = (Base * Base) rem Mod,
+    mod_pow(Base2, Exp div 2, Mod, Acc1).
+
+%% Modular multiplication
+mod_mul(A, B, Mod) ->
+    (A * B) rem Mod.
+
+%% Test cases
+task(TestCases) ->
+    lists:foreach(fun([N, P]) ->
+        try
+            R = tonelli(N, P),
+            io:format("n = ~p p = ~p~n  roots : ~p ~p~n", [N, P, R, P - R])
+        catch
+            _:Error ->
+                io:format("Error for n=~p, p=~p: ~p~n", [N, P, Error])
+        end
+    end, TestCases).
+
+main(_) ->
+    Cases = [
+        [10, 13],
+        [56, 101],
+        [1030, 10009],
+        [44402, 100049],
+        [665820697, 1000000009],
+        [881398088036, 1000000000039],
+        [41660815127637347468140745042827704103445750172002, 100000000000000000000000000000000000000000000000577]
+    ],
+    task(Cases),
+
+    % Test error case
+    try
+        tonelli(1032, 1009),
+        io:format("Error: Should have failed~n")
+    catch
+        _:Error ->
+            io:format("Correctly caught error: ~p~n", [Error])
+    end.