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|
-- This program is free software; you can redistribute it and/or
-- modify it under the terms of the GNU General Public License as
-- published by the Free Software Foundation; either version 2 of the
-- License, or (at your option) any later version.
-- This program is distributed in the hope that it will be useful,
-- but WITHOUT ANY WARRANTY; without even the implied warranty of
-- MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
-- General Public License for more details.
-- You should have received a copy of the GNU General Public License
-- along with this program; if not, write to the Free Software
-- Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA
-- 02111-1307, USA.
-- As a special exception, if other files instantiate generics from
-- this unit, or you link this unit with other files to produce an
-- executable, this unit does not by itself cause the resulting
-- executable to be covered by the GNU General Public License. This
-- exception does not however invalidate any other reasons why the
-- executable file might be covered by the GNU Public License.
with Crypto.Types.Random;
with Crypto.Asymmetric.Prime_Tables;
--with Ada.Integer_Text_IO; use Ada.Integer_Text_IO;
with Ada.Text_IO;
separate(Crypto.Types.Big_Numbers)
package body Mod_Utils is
pragma Optimize (Time);
use Crypto.Asymmetric.Prime_Tables;
---------------------------------------------------------------------------
function Patch(Item, N : Big_Unsigned)
return Big_Unsigned is
Diff : constant Big_Unsigned:=((Big_Unsigned_Last - N) + 1) mod N;
begin
return Add(Item,Diff,N);
end Patch; pragma Inline(Patch);
---------------------------------------------------------------------------
function Add(Left, Right, N : Big_Unsigned) return Big_Unsigned is
L : constant Big_Unsigned := Left mod N;
R : constant Big_Unsigned := Right mod N;
Result : constant Big_Unsigned := L + R;
begin
if Result < Max(L,R) then
return Patch(Result,N);
else return
Result mod N;
end if;
end Add;
---------------------------------------------------------------------------
function Sub(Left, Right, N : Big_Unsigned) return Big_Unsigned is
L : constant Big_Unsigned := Left mod N;
R : constant Big_Unsigned := Right mod N;
begin
if R > L then
return N - R + L;
else return L-R;
end if;
end Sub;
---------------------------------------------------------------------------
function Div(Left, Right, N : Big_Unsigned) return Big_Unsigned is
begin
return Mult(Left,Inverse(Right,N),N);
end Div; pragma Inline(Div);
---------------------------------------------------------------------------
--from Erik-Zenners handout "Zahlentheoretische Algorithmen"
function Pow(Base, Exponent, N : Big_Unsigned) return Big_Unsigned is
L : constant Big_Unsigned := Base mod N;
R : constant Big_Unsigned := Exponent;
Result : Big_Unsigned := Big_Unsigned_One;
begin
if L = Big_Unsigned_Zero or L = Big_Unsigned_One then
return L;
elsif R = Big_Unsigned_Zero then return Big_Unsigned_One;
else
-- Square_And_Muliply
for I in reverse 0..Bit_Length(R)-1 loop
Result := Mult(Result,Result,N);
if (Shift_Right(R, I) mod 2) = Big_Unsigned_One then
Result := Mult(Result,L,N);
end if;
end loop;
return Result mod N;
end if;
end Pow;
---------------------------------------------------------------------------
--based on Erik-Zenners handout "Zahlentheoretische Algorithmen"
-- (ext. Euklid)
-- This function returns Big_unsigned_Zero if X have no inverse mod n
function Inverse(X, N : Big_Unsigned) return Big_Unsigned is
B : Big_Unsigned := X mod N;
A : Big_Unsigned := N;
begin
-- if gcd(A,B) /= 1 then A have no inverse mod B
if B = Big_Unsigned_Zero or A = Big_Unsigned_Zero or
Gcd(A,B) /= Big_Unsigned_One then
return Big_Unsigned_Zero;
end if;
declare
T : Big_Unsigned := Big_Unsigned_One;
Tstrich, Tempt : Big_Unsigned;
Q, R : Big_Unsigned;
begin
loop
Big_Div(A,B,Q,R);
if(R = Big_Unsigned_Zero) then
return T;
end if;
A:=B;
B:=R;
Tempt:=T;
T:=Sub(Tstrich,Mult(Q,T,N),N);
Tstrich:=Tempt;
end loop;
end;
end Inverse;
---------------------------------------------------------------------------
function Get_Random(N : Big_Unsigned) return Big_Unsigned is
Result : Big_Unsigned;
begin
Random.Read(Result.Number);
for I in reverse 0..N.Last_Index loop
if Result.Number(I) /= 0 then
Result.Last_Index := I;
exit;
end if;
end loop;
return Result mod N ;
end Get_Random;
---------------------------------------------------------------------------
-- this function returns true if X is a Mersenne prim number
function Lucas_Lehmer_Test(X : Big_Unsigned) return Boolean is
Is_Mp : Boolean := false;
begin
if X.Last_Index = 0 then
for I in 2..Word'Size-1 loop
if X.Number(0) = Shift_Left(2,I)-1 then
Is_Mp := True;
exit;
end if;
end loop;
if Is_Mp = False then return False;
end if;
else
for I in 0..X.Last_Index loop
if X.Number(I) /= Word'Last then return False;
end if;
end loop;
end if;
declare
P : constant Word := Word(Bit_Length(X)-1);
S : Big_Unsigned := Big_Unsigned_Two+2; --S(1) = 4;
begin
for I in 2..P-1 loop
S := (Mult(S,S,X) - 2) mod X;
end loop;
if S = Big_Unsigned_Zero then return True;
else return False;
end if;
end;
end Lucas_Lehmer_Test;
---------------------------------------------------------------------------
--from Erik-Zenners handout "Zahlentheoretische Algorithmen"
function Is_Miller_Rabin_Witness(Wit, X : Big_Unsigned) return Boolean is
B : constant Big_Unsigned := X-1;
Result : Big_Unsigned := Big_Unsigned_One;
Root : Big_Unsigned;
begin
for I in reverse 0..Bit_Length(B)-1 loop
Root := Result;
Result := Mult(Result, Result, X);
if ((Result = Big_Unsigned_One) and
(Root /= Big_Unsigned_One and Root /= B)) then return True;
elsif (Shift_Right(B,I) mod 2) = Big_Unsigned_One then
Result := Mult(Result, Wit, X);
end if;
end loop;
if Result /= Big_Unsigned_One then return True;
else return False;
end if;
end Is_Miller_Rabin_Witness;
---------------------------------------------------------------------------
-- Test if Wit is a witness for N
-- If Wit is a wittness then N is no prime
function Is_Simple_Witness(Wit, N : Big_Unsigned) return Boolean is
begin
-- is Wit a "Miller-Rabin"-witness
if (Wit /= (N-Big_Unsigned_One)) and (Wit /= Big_Unsigned_One) and
Mult(Wit,Wit,N) = Big_Unsigned_One then return True;
elsif Gcd(Wit,N) /= Big_Unsigned_One then return True;
-- is Wit a "Fermat-Witness"
-- elsif Pow(Wit,N-1,N) /= Big_Unsigned_One then return True;
else return False;
end if;
end Is_Simple_Witness;
---------------------------------------------------------------------------
-- Returns true if N passes the specified number of Miller-Rabin tests.
function Passed_Miller_Rabin_Test(X : Big_Unsigned; S : Positive)
return Boolean is
Witness : Big_Unsigned;
begin
-- Do the tests
for I in 1..S loop
-- Generate a uniform random on (1, X)
loop
Witness := Get_Random(X);
exit when Witness > Big_Unsigned_One;
end loop;
if Is_Miller_Rabin_Witness(Witness, X) then
return False;
end if;
end loop;
return true;
end Passed_Miller_Rabin_Test;
---------------------------------------------------------------------------
function Pass_Prime_Test(X : Big_Unsigned; Status : Hardness)
return Boolean is
Rounds : Natural;
X_Bit_Size : constant Natural := Bit_Length(X);
begin
if X < Big_Unsigned_Two then return False;
elsif Is_Even(X) then
if X = Big_Unsigned_Two then return True;
else return False;
end if;
end if;
--X is odd
for I in One_Digit_Primes'First+1..One_Digit_Primes'Last loop
if X = Word(One_Digit_Primes(I)) then return true;
elsif X mod Word(One_Digit_Primes(I)) = Big_Unsigned_Zero then
return False;
end if;
end loop;
for I in Two_Digit_Primes'Range loop
if X = Word(Two_Digit_Primes(I)) then return true;
elsif X mod Word(Two_Digit_Primes(I)) = Big_Unsigned_Zero then
return False;
end if;
end loop;
if Lucas_Lehmer_Test(X) then
return True;
end if;
for I in Three_Digit_Primes'Range loop
if X = Word(Three_Digit_Primes(I)) then return true;
elsif X mod Word(Three_Digit_Primes(I)) = Big_Unsigned_Zero then
return False;
end if;
end loop;
-- The relationship between the certainty and the number of rounds
-- we perform is given in the draft standard ANSI X9.80, "PRIME
-- NUMBER GENERATION, PRIMALITY TESTING, AND PRIMALITY CERTIFICATES".
-- Comment:
-- I don't have a look on this paper. =:) I borrowed this
-- "algorithmen" from the j2sdk1.4.1 library (java/math/BigInteger.java)
-- If you have the permission to send me the draft standard ANSI X9.80
-- then send it, please!
-- I'm a student. I have no money for ANSI or IEEE drafts. :-(
-- It's right to require money to read a draft?
-- This really really sucks! SCNR!
if (X_Bit_Size < 100) then Rounds := 50;
elsif (X_Bit_Size < 256) then Rounds := 27;
elsif (X_Bit_Size < 512) then Rounds := 15;
elsif (X_Bit_Size < 768) then Rounds := 8;
elsif (X_Bit_Size < 1024) then Rounds := 4;
else Rounds := 2;
end if;
declare
Witness : Big_Unsigned;
begin
if Status = Weak then
for I in 1..Rounds loop
loop
Witness := Get_Random(X);
exit when Witness > Big_Unsigned_Two;
end loop;
if Is_Simple_Witness(Witness,X) then return False;
end if;
end loop;
else
for I in 1..Rounds loop
loop
Witness := Get_Random(X);
exit when Witness > Big_Unsigned_Two;
end loop;
if Is_Miller_Rabin_Witness(Witness,X) then return False;
end if;
end loop;
end if;
end;
return True;
end Pass_Prime_Test;
---------------------------------------------------------------------------
function Is_Prime(X : Big_Unsigned) return Boolean is
begin
return Pass_Prime_Test(X, Strong);
end Is_Prime; pragma Inline (Is_Prime);
---------------------------------------------------------------------------
-- This function is faster then Is_prime but a lot of no strong pseudo
-- primes pass this test
function Looks_Like_A_Prime(X : Big_Unsigned) return Boolean is
begin
return Pass_Prime_Test(X, Weak);
end Looks_Like_A_Prime; pragma Inline(Looks_Like_A_Prime);
---------------------------------------------------------------------------
function Get_Prime(N : Big_Unsigned) return Big_Unsigned is
Result : Big_Unsigned := Get_Random(N);
begin
if N <= Big_Unsigned_Two then
raise Constraint_Error;
end if;
-- make sure that Result is odd
Result.Number(0) := Result.Number(0) or 1;
loop
if Is_Prime(Result) then return Result;
else Result := (Result+2) mod N ;
end if;
end loop;
end Get_Prime;
---------------------------------------------------------------------------
function "mod"(Left : D_Big_Unsigned; Right : Big_Unsigned)
return Big_Unsigned;
---------------------------------------------------------------------------
-- Result = Left * Right (mod N)
function Mult(Left, Right, N : Big_Unsigned) return Big_Unsigned is
T : DWord;
Carry : Word := 0;
R : D_Big_Unsigned;
begin
for I in 0..Left.Last_Index loop
for J in 0..Right.Last_Index loop
T := DWord(Left.Number(I)) * DWord(Right.Number(J))
+ DWord(R.Number(I+J)) + DWord(Carry);
R.Number(I+J) := Word(T and DWord(Word'Last));
Carry:= Word(Shift_Right(T,Word'Size));
end loop;
R.Number(I+Right.Last_Index+1) := Carry +
R.Number(I+Right.Last_Index+1);
Carry := 0;
end loop;
for I in reverse 0..D_Max_Length loop
if R.Number(I) /= 0 then
R.Last_Index := I;
exit;
end if;
end loop;
return R mod N;
end Mult;
---------------------------------------------------------------------------
-- Returns a probability N-bit prime (Result).
function Get_N_Bit_Prime(N : Positive) return Big_Unsigned is
J : Big_Unsigned := Get_Random(Shift_Left(Big_Unsigned_One,N-2));
Index : constant Natural := (N-1)/Word'Size;
Amount : constant Natural := (N-1) mod Word'Size;
Result : Big_Unsigned := Shift_Left(J,1);
begin
if N = 1 or N > Size then
raise Constraint_Error;
end if;
loop
-- Make sure that Result is an odd
Set_Least_Significant_Bit (Result);
-- Make sure that Result is a N-Bit-Number;
Result.Number (Index) := Result.Number (Index) or
Shift_Left (Word (1), Amount);
if Amount = 0 then
Result.Last_Index := Index;
end if;
if Is_Prime(Result) then
return Result;
else
Result:=Result-2;
if Is_Prime(Result) then
return Result;
end if;
end if;
J := Get_Random (Shift_Left (Big_Unsigned_One, N - 2));
Result := Shift_Left (J, 1);
end loop;
end Get_N_Bit_Prime;
---------------------------------------------------------------------------
-- computes the jacobi-symbol
-- return value:
-- 0 : if X mod N = 0
-- 1 : if X is a quadratic resuide mod N
-- -1 : if X is a quadratic non-resuide mod N
function Jacobi(X, N : Big_Unsigned) return Integer is
A : Big_Unsigned := X mod N;
begin
if Is_Even(N) then
raise Constraint_Error;
end if;
if N = Big_Unsigned_One then return 1;
elsif A = Big_Unsigned_Zero then return 0;
elsif A = Big_Unsigned_One then return 1;
end if;
while (A mod 4) = Big_Unsigned_Zero loop
exit when (A mod 4) = Big_Unsigned_Zero;
A := Shift_Right(A,2);
end loop;
if Is_Even(A) then
if (N mod 8 = 1) or (N mod 8 = 7) then
return Jacobi(Shift_Right(A,1),N);
else return -1*Jacobi(Shift_Right(A,1),N);
end if;
else
if (A mod 4 = 1) or (N mod 4 = 1) then
return Jacobi(N mod A, A);
else return -1*Jacobi(N mod A, A);
end if;
end if;
end Jacobi;
----------------------------------------------------------------------------
-----------------------------DOUBLE_SIZE------------------------------------
----------------------------------------------------------------------------
--only needed for multiplication mod N
--here we need 2*Size-bit numbers to avoid an overflow because
--if one of our provisional result t > BIG_Unsigned_Last
--then there ist no well known algortihm to compute the
-- result of an multiplication mod m
-- same algorithm for D_Big_Unsigned as for Big_Unsigned
function "="(Left, Right : D_Big_Unsigned) return Boolean is
begin
if Left.Last_Index = Right.Last_Index then
for I in 0..Left.Last_Index loop
if Left.Number(I) /= Right.Number(I) then return False;
end if;
end loop;
else return False;
end if;
return True;
end"=";
----------------------------------------------------------------------------
function Shift_Left(Value : D_Big_Unsigned; Amount : Natural)
return D_Big_Unsigned is
begin
if Amount >= (D_Max_Length+1)*Word'Size or
Value = D_Big_Unsigned_Zero
then return D_Big_Unsigned_Zero;
elsif Amount = 0 then return Value;
end if;
declare
Result : D_Big_Unsigned;
Temp : DLimbs :=(others => 0);
L : constant Natural := Amount mod Word'Size;
R : constant Natural := Word'Size-L;
M : constant Natural := Amount/Word'Size;
begin
Temp(0) := Shift_Left(Value.Number(0), L);
for I in 1..Value.Last_Index loop
Temp(I) := Shift_Right(Value.Number(I-1), R) +
Shift_Left(Value.Number(I), L);
end loop;
if Value.Last_Index /= D_Max_Length then
Temp(Value.Last_Index+1):=
Shift_Right(Value.Number(Value.Last_Index), R);
end if;
for I in Temp'Range loop
if (I+M) > D_Max_Length then
exit;
end if;
Result.Number(I+M):= Temp(I);
end loop;
for I in reverse 0..D_Max_Length loop
if Result.Number(I) /=0 then
Result.Last_Index:=I;
exit;
end if;
end loop;
return Result;
end;
end Shift_Left; pragma Inline (Shift_Left);
---------------------------------------------------------------------------
function Bit_Length(X : D_Big_Unsigned) return Natural is
begin
if X = D_Big_Unsigned_Zero then
return 0;
end if;
for I in reverse 0..Word'Size-1 loop
if Shift_Left(1,I) <= X.Number(X.Last_Index) then
return Word'Size * X.Last_Index + I + 1 ;
end if;
end loop;
return X.Last_Index * Word'Size;
end Bit_Length; pragma Inline (Bit_Length);
---------------------------------------------------------------------------
function "<"(Left, Right : D_Big_Unsigned) return Boolean is
begin
if Left.Last_Index < Right.Last_Index then return True;
elsif Left.Last_Index > Right.Last_Index then return False;
else
for I in reverse 0..Left.Last_Index loop
if Left.Number(I) < Right.Number(I) then return True;
elsif Left.Number(I) > Right.Number(I) then return False;
end if;
end loop;
end if;
return False;
end "<"; pragma Inline ("<");
---------------------------------------------------------------------------
function ">"(Left, Right : D_Big_Unsigned) return Boolean is
begin
return Right < Left;
end ">"; pragma Inline (">");
---------------------------------------------------------------------------
function ">="(Left, Right : D_Big_Unsigned) return Boolean is
begin
return not(Left < Right);
end ">="; pragma Inline (">=");
---------------------------------------------------------------------------
function "+"(Left, Right : D_Big_Unsigned) return D_Big_Unsigned;
function "-"(Left, Right : D_Big_Unsigned) return D_Big_Unsigned is
begin
if Left = Right then return D_Big_Unsigned_Zero;
elsif Left = Right+D_Big_Unsigned_One then return D_Big_Unsigned_One;
elsif Left+D_Big_Unsigned_One = Right then return D_Big_Unsigned_Last;
-- add the modulus if Right > Left
elsif Right > Left then
return D_Big_Unsigned_Last - Right + Left + D_Big_Unsigned_One;
else
declare
Result : D_Big_Unsigned;
Carry : Word:=0;
begin
-- Remember Left > Right
for I in 0..Left.Last_Index loop
Result.Number(I) := Left.Number(I) - Right.Number(I) - Carry;
if (Right.Number(I) > Left.Number(I)) or
(Carry= 1 and Right.Number(I) = Left.Number(I))
then Carry :=1;
else Carry :=0;
end if;
if Result.Number(I) /= 0 then
Result.Last_Index := I;
end if;
end loop;
return Result;
end;
end if;
end "-";
---------------------------------------------------------------------------
function "mod"(Left : D_Big_Unsigned; Right : Big_Unsigned)
return Big_Unsigned is
begin
if Left.Last_Index <= Max_Length then
declare
L : Big_Unsigned;
begin
L.Last_Index := Left.Last_Index;
L.Number(0..Left.Last_Index) := Left.Number(0..Left.Last_Index);
return L mod Right;
end;
end if;
if Right = Big_Unsigned_Zero then raise Division_By_Zero;
--elsif Right = Big_Unsigned_One then return Big_Unsigned_Zero;
end if;
-- Now, there is only the case where (Left > Right), (Right /= 0)
-- and |Left|>|Right|.
declare
Remainder : D_Big_Unsigned:=Left;
Temp_Right, R : D_Big_Unsigned;
Result : Big_Unsigned;
Diff: Natural;
begin
Temp_Right.Last_Index := Right.Last_Index;
Temp_Right.Number(0..Right.Last_Index) :=
Right.Number(0..Right.Last_Index);
R:=Temp_Right;
while(Remainder >= R) loop
Diff := Bit_Length(Remainder) - Bit_Length(R);
if Diff = 0 then
Remainder := Remainder-R;
exit;
else Diff:=Diff-1;
end if;
Temp_Right := Shift_Left(R, Diff);
Remainder := Remainder-Temp_Right;
end loop;
Result.Last_Index := Remainder.Last_Index;
Result.Number(0..Result.Last_Index) :=
Remainder.Number(0..Result.Last_Index);
return Result;
end;
end "mod";
---------------------------------------------------------------------------
function "+"(Left, Right : D_Big_Unsigned) return D_Big_Unsigned is
Result : D_Big_Unsigned;
M : constant Natural := Natural'Max(Left.Last_Index, Right.Last_Index);
Temp : Word;
Carry : Word :=0;
begin
for I in 0..M loop
Temp :=Carry;
Result.Number(I) := Left.Number(I) + Right.Number(I) +Temp;
if Result.Number(I) < Word'Max(Left.Number(I), Right.Number(I))
then Carry := 1;
else Carry := 0;
end if;
end loop;
if Carry =1 and M < Max_Length then
Result.Number(M+1) := 1;
Result.Last_Index := M+1;
else
-- Set Result.Last_Index
for I in reverse 0..M loop
if Result.Number(I) /= 0 then
Result.Last_Index := I;
return Result;
end if;
end loop;
end if;
return Result;
end "+";
---------------------------------------------------------------------------
end Mod_Utils;
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