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Ingy döt Net
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' version 11-04-2017
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' compile with: fbc -s console
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' maximum for p is 17 digits to be on the save side
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' TRUE/FALSE are built-in constants since FreeBASIC 1.04
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' But we have to define them for older versions.
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#Ifndef TRUE
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#Define FALSE 0
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#Define TRUE Not FALSE
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#EndIf
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Function mul_mod(a As ULongInt, b As ULongInt, modulus As ULongInt) As ULongInt
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' returns a * b mod modulus
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Dim As ULongInt x, y = a Mod modulus
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While b > 0
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If (b And 1) = 1 Then
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x = (x + y) Mod modulus
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End If
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y = (y Shl 1) Mod modulus
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b = b Shr 1
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Wend
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Return x
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End Function
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Function pow_mod(b As ULongInt, power As ULongInt, modulus As ULongInt) As ULongInt
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' returns b ^ power mod modulus
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Dim As ULongInt x = 1
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While power > 0
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If (power And 1) = 1 Then
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' x = (x * b) Mod modulus
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x = mul_mod(x, b, modulus)
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End If
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' b = (b * b) Mod modulus
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b = mul_mod(b, b, modulus)
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power = power Shr 1
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Wend
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Return x
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End Function
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Function Isprime(n As ULongInt, k As Long) As Long
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' miller-rabin prime test
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If n > 9223372036854775808ull Then ' limit 2^63, pow_mod/mul_mod can't handle bigger numbers
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Print "number is to big, program will end"
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Sleep
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End
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End If
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' 2 is a prime, if n is smaller then 2 or n is even then n = composite
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If n = 2 Then Return TRUE
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If (n < 2) OrElse ((n And 1) = 0) Then Return FALSE
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Dim As ULongInt a, x, n_one = n - 1, d = n_one
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Dim As UInteger s
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While (d And 1) = 0
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d = d Shr 1
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s = s + 1
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Wend
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While k > 0
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k = k - 1
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a = Int(Rnd * (n -2)) +2 ' 2 <= a < n
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x = pow_mod(a, d, n)
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If (x = 1) Or (x = n_one) Then Continue While
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For r As Integer = 1 To s -1
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x = pow_mod(x, 2, n)
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If x = 1 Then Return FALSE
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If x = n_one Then Continue While
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Next
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If x <> n_one Then Return FALSE
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Wend
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Return TRUE
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End Function
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Function legendre_symbol (a As LongInt, p As LongInt) As LongInt
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Dim As LongInt x = pow_mod(a, ((p -1) \ 2), p)
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If p -1 = x Then
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Return x - p
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Else
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Return x
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End If
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End Function
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' ------=< MAIN >=------
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Dim As LongInt b, c, i, k, m, n, p, q, r, s, t, z
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For k = 1 To 7
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Read n, p
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Print "Find solution for n ="; n; " and p =";p
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If legendre_symbol(n, p) <> 1 Then
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Print n;" is not a quadratic residue"
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Print
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Continue For
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End If
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If p = 2 OrElse Isprime(p, 15) = FALSE Then
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Print p;" is not a odd prime"
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Print
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Continue For
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End If
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s = 0 : q = p -1
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Do
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s += 1
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q \= 2
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Loop Until (q And 1) = 1
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If s = 1 And (p Mod 4) = 3 Then
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r = pow_mod(n, ((p +1) \ 4), p)
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Print "Solution found:"; r; " and"; p - r
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Print
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Continue For
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End If
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z = 1
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Do
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z += 1
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Loop Until legendre_symbol(z, p) = -1
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c = pow_mod(z, q, p)
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r = pow_mod(n, (q +1) \ 2, p)
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t = pow_mod(n, q, p)
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m = s
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Do
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i = 0
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If (t Mod p) = 1 Then
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Print "Solution found:"; r; " and"; p - r
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Print
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Continue For
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End If
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Do
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i += 1
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If i >= m Then Continue For
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Loop Until pow_mod(t, 2 ^ i, p) = 1
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b = pow_mod(c, (2 ^ (m - i -1)), p)
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r = mul_mod(r, b, p)
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c = mul_mod(b, b, p)
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t = mul_mod(t, c, p)' t = t * b ^ 2
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m = i
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Loop
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Next
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Data 10, 13, 56, 101, 1030, 10009, 1032, 10009, 44402, 100049
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Data 665820697, 1000000009, 881398088036, 1000000000039
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' empty keyboard buffer
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While InKey <> "" : Wend
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Print : Print "hit any key to end program"
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Sleep
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End
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@ -0,0 +1,131 @@
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' version 12-04-2017
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' compile with: fbc -s console
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#Include Once "gmp.bi"
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Data "10", "13", "56", "101", "1030", "10009", "1032", "10009"
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Data "44402", "100049", "665820697", "1000000009"
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Data "881398088036", "1000000000039"
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Data "41660815127637347468140745042827704103445750172002" ' p = 10^50 + 577
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' ------=< MAIN >=------
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Dim As uLong k
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Dim As ZString Ptr zstr
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Dim As String n_str, p_str
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Dim As Mpz_ptr b, c, i, m, n, p, q, r, s, t, z, tmp
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b = Allocate(Len(__Mpz_struct)) : Mpz_init(b)
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c = Allocate(Len(__Mpz_struct)) : Mpz_init(c)
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i = Allocate(Len(__Mpz_struct)) : Mpz_init(i)
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m = Allocate(Len(__Mpz_struct)) : Mpz_init(m)
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n = Allocate(Len(__Mpz_struct)) : Mpz_init(n)
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p = Allocate(Len(__Mpz_struct)) : Mpz_init(p)
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q = Allocate(Len(__Mpz_struct)) : Mpz_init(q)
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r = Allocate(Len(__Mpz_struct)) : Mpz_init(r)
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s = Allocate(Len(__Mpz_struct)) : Mpz_init(s)
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t = Allocate(Len(__Mpz_struct)) : Mpz_init(t)
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z = Allocate(Len(__Mpz_struct)) : Mpz_init(z)
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tmp = Allocate(Len(__Mpz_struct)) : Mpz_init(tmp)
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For k = 1 To 8
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Read n_str
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Mpz_set_str(n, n_str, 10)
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If k < 8 Then
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Read p_str
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Mpz_set_str(p, p_str, 10)
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Else
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p_str = "10^50 + 577"
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Mpz_set_str(p, "1" + String(50, "0"), 10)
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Mpz_add_ui(p, p, 577)
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End If
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Print "Find solution for n = "; n_str; " and p = "; p_str
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If Mpz_legendre(n, p) <> 1 Then
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Print n_str; " is not a quadratic residue"
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Print
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Continue For
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End If
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If Mpz_tstbit(p, 0) = 0 OrElse Mpz_probab_prime_p(p, 20) = 0 Then
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Print p_str; "is not a odd prime"
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Print
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Continue For
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End If
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Mpz_set_ui(s, 0) : Mpz_set(q, p) : Mpz_sub_ui(q, q, 1) ' q = p -1
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Do
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Mpz_add_ui(s, s, 1)
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Mpz_fdiv_q_2exp(q, q, 1)
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Loop Until Mpz_tstbit(q, 0) = 1
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If Mpz_cmp_ui(s, 1) = 0 Then
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If Mpz_tstbit(p, 1) = 1 Then
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Mpz_add_ui(tmp, p, 1)
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Mpz_fdiv_q_2exp(tmp, tmp, 2) ' tmp = p +1 \ 4
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Mpz_powm(r, n, tmp, p)
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zstr = Mpz_get_str(0, 10, r)
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Print "Solution found: "; *zstr;
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Mpz_sub(r, p, r)
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zstr = Mpz_get_str(0, 10, r)
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Print " and "; *zstr
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Print
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Continue For
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End If
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End If
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Mpz_set_ui(z, 1)
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Do
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Mpz_add_ui(z, z, 1)
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Loop Until Mpz_legendre(z, p) = -1
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Mpz_powm(c, z, q, p)
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Mpz_add_ui(tmp, q, 1)
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Mpz_fdiv_q_2exp(tmp, tmp, 1)
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Mpz_powm(r, n, tmp, p)
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Mpz_powm(t, n, q, p)
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Mpz_set(m, s)
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Do
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Mpz_set_ui(i, 0)
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Mpz_mod(tmp, t, p)
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If Mpz_cmp_ui(tmp, 1) = 0 Then
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zstr = Mpz_get_str(0, 10, r)
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Print "Solution found: "; *zstr;
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Mpz_sub(r, p, r)
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zstr = Mpz_get_str(0, 10, r)
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Print " and "; *zstr
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Print
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Continue For
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End If
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Mpz_set_ui(q, 1)
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Do
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Mpz_add_ui(i, i, 1)
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If Mpz_cmp(i, m) >= 0 Then
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Continue For
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end if
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Mpz_mul_ui(q, q, 2) ' q = 2^i
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Mpz_powm(tmp, t, q, p)
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Loop Until Mpz_cmp_ui(tmp, 1) = 0
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Mpz_set_ui(q, 2)
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Mpz_sub(tmp, m, i) : Mpz_sub_ui(tmp, tmp, 1) : Mpz_powm(tmp, q, tmp, p)
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Mpz_powm(b, c, tmp, p)
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Mpz_mul(r, r, b) : Mpz_mod(r, r, p)
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Mpz_mul(tmp, b, b) : Mpz_mod(c, tmp, p)
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Mpz_mul(tmp, t, c) : Mpz_mod(t, tmp, p)
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Mpz_set(m, i)
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Loop
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Next
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Mpz_clear(b) : Mpz_clear(c) : Mpz_clear(i) : Mpz_clear(m)
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Mpz_clear(n) : Mpz_clear(p) : Mpz_clear(q) : Mpz_clear(r)
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Mpz_clear(s) : Mpz_clear(t) : Mpz_clear(z) : Mpz_clear(tmp)
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' empty keyboard buffer
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While InKey <> "" : Wend
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Print : Print "hit any key to end program"
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Sleep
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End
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