Implementierung des Diffie-Hellman-Algorithmus
Diffie-Hellman-Algorithmus:
Der Diffie-Hellman-Algorithmus wird verwendet, um ein gemeinsames Geheimnis zu etablieren, das für geheime Kommunikationen beim Datenaustausch über ein öffentliches Netzwerk verwendet werden kann. Dabei wird die elliptische Kurve verwendet, um Punkte zu generieren und mithilfe der Parameter den geheimen Schlüssel zu erhalten.
- Der Einfachheit halber und der praktischen Umsetzung des Algorithmus halber betrachten wir nur vier Variablen, eine Primzahl P und G (eine Primitivwurzel von P) und zwei private Werte a und b.
- P und G sind beide öffentlich verfügbare Nummern. Benutzer (z. B. Alice und Bob) wählen die privaten Werte a und b aus, generieren einen Schlüssel und tauschen ihn öffentlich aus. Die Gegenperson erhält den Schlüssel und dieser generiert einen geheimen Schlüssel, nach dem sie denselben geheimen Schlüssel verschlüsseln muss.
Die Schritt-für-Schritt-Erklärung lautet wie folgt:
| Alice | Bob |
|---|---|
| Öffentliche Schlüssel verfügbar = P G | Öffentliche Schlüssel verfügbar = P G |
| Ausgewählter privater Schlüssel = a | Ausgewählter privater Schlüssel = b |
| Schlüssel generiert = x = G^a mod P | Schlüssel generiert = y = G^b mod P |
| Es findet ein Austausch der generierten Schlüssel statt | |
| Schlüssel erhalten = y | Schlüssel erhalten = x |
| Generierter geheimer Schlüssel = k_a = y^a mod P | Generierter geheimer Schlüssel = k_b = x^b mod P |
| Algebraisch lässt sich das zeigen k_a = k_b | |
| Benutzer verfügen nun über einen symmetrischen geheimen Schlüssel zum Verschlüsseln |
Beispiel:
Step 1: Alice and Bob get public numbers P = 23 G = 9
Step 2: Alice selected a private key a = 4 and
Bob selected a private key b = 3
Step 3: Alice and Bob compute public values
Alice: x =(9^4 mod 23) = (6561 mod 23) = 6
Bob: y = (9^3 mod 23) = (729 mod 23) = 16
Step 4: Alice and Bob exchange public numbers
Step 5: Alice receives public key y =16 and
Bob receives public key x = 6
Step 6: Alice and Bob compute symmetric keys
Alice: ka = y^a mod p = 65536 mod 23 = 9
Bob: kb = x^b mod p = 216 mod 23 = 9
Step 7: 9 is the shared secret.
Durchführung:
C++C/* This program calculates the Key for two persons using the Diffie-Hellman Key exchange algorithm using C++ */ #include#include using namespace std ; // Power function to return value of a ^ b mod P long long int power ( long long int a long long int b long long int P ) { if ( b == 1 ) return a ; else return ((( long long int ) pow ( a b )) % P ); } // Driver program int main () { long long int P G x a y b ka kb ; // Both the persons will be agreed upon the // public keys G and P P = 23 ; // A prime number P is taken cout < < 'The value of P : ' < < P < < endl ; G = 9 ; // A primitive root for P G is taken cout < < 'The value of G : ' < < G < < endl ; // Alice will choose the private key a a = 4 ; // a is the chosen private key cout < < 'The private key a for Alice : ' < < a < < endl ; x = power ( G a P ); // gets the generated key // Bob will choose the private key b b = 3 ; // b is the chosen private key cout < < 'The private key b for Bob : ' < < b < < endl ; y = power ( G b P ); // gets the generated key // Generating the secret key after the exchange // of keys ka = power ( y a P ); // Secret key for Alice kb = power ( x b P ); // Secret key for Bob cout < < 'Secret key for the Alice is : ' < < ka < < endl ; cout < < 'Secret key for the Bob is : ' < < kb < < endl ; return 0 ; } // This code is contributed by Pranay Arora Java/* This program calculates the Key for two persons using the Diffie-Hellman Key exchange algorithm */ #include#include // Power function to return value of a ^ b mod P long long int power ( long long int a long long int b long long int P ) { if ( b == 1 ) return a ; else return ((( long long int ) pow ( a b )) % P ); } // Driver program int main () { long long int P G x a y b ka kb ; // Both the persons will be agreed upon the // public keys G and P P = 23 ; // A prime number P is taken printf ( 'The value of P : %lld n ' P ); G = 9 ; // A primitive root for P G is taken printf ( 'The value of G : %lld nn ' G ); // Alice will choose the private key a a = 4 ; // a is the chosen private key printf ( 'The private key a for Alice : %lld n ' a ); x = power ( G a P ); // gets the generated key // Bob will choose the private key b b = 3 ; // b is the chosen private key printf ( 'The private key b for Bob : %lld nn ' b ); y = power ( G b P ); // gets the generated key // Generating the secret key after the exchange // of keys ka = power ( y a P ); // Secret key for Alice kb = power ( x b P ); // Secret key for Bob printf ( 'Secret key for the Alice is : %lld n ' ka ); printf ( 'Secret Key for the Bob is : %lld n ' kb ); return 0 ; } Python// This program calculates the Key for two persons // using the Diffie-Hellman Key exchange algorithm class GFG { // Power function to return value of a ^ b mod P private static long power ( long a long b long p ) { if ( b == 1 ) return a ; else return ((( long ) Math . pow ( a b )) % p ); } // Driver code public static void main ( String [] args ) { long P G x a y b ka kb ; // Both the persons will be agreed upon the // public keys G and P // A prime number P is taken P = 23 ; System . out . println ( 'The value of P:' + P ); // A primitive root for P G is taken G = 9 ; System . out . println ( 'The value of G:' + G ); // Alice will choose the private key a // a is the chosen private key a = 4 ; System . out . println ( 'The private key a for Alice:' + a ); // Gets the generated key x = power ( G a P ); // Bob will choose the private key b // b is the chosen private key b = 3 ; System . out . println ( 'The private key b for Bob:' + b ); // Gets the generated key y = power ( G b P ); // Generating the secret key after the exchange // of keys ka = power ( y a P ); // Secret key for Alice kb = power ( x b P ); // Secret key for Bob System . out . println ( 'Secret key for the Alice is:' + ka ); System . out . println ( 'Secret key for the Bob is:' + kb ); } } // This code is contributed by raghav14C## Diffie-Hellman Code # Power function to return value of a^b mod P def power ( a b p ): if b == 1 : return a else : return pow ( a b ) % p # Main function def main (): # Both persons agree upon the public keys G and P # A prime number P is taken P = 23 print ( 'The value of P:' P ) # A primitive root for P G is taken G = 9 print ( 'The value of G:' G ) # Alice chooses the private key a # a is the chosen private key a = 4 print ( 'The private key a for Alice:' a ) # Gets the generated key x = power ( G a P ) # Bob chooses the private key b # b is the chosen private key b = 3 print ( 'The private key b for Bob:' b ) # Gets the generated key y = power ( G b P ) # Generating the secret key after the exchange of keys ka = power ( y a P ) # Secret key for Alice kb = power ( x b P ) # Secret key for Bob print ( 'Secret key for Alice is:' ka ) print ( 'Secret key for Bob is:' kb ) if __name__ == '__main__' : main ()JavaScript// C# implementation to calculate the Key for two persons // using the Diffie-Hellman Key exchange algorithm using System ; class GFG { // Power function to return value of a ^ b mod P private static long power ( long a long b long P ) { if ( b == 1 ) return a ; else return ((( long ) Math . Pow ( a b )) % P ); } public static void Main () { long P G x a y b ka kb ; // Both the persons will be agreed upon the // public keys G and P P = 23 ; // A prime number P is taken Console . WriteLine ( 'The value of P:' + P ); G = 9 ; // A primitive root for P G is taken Console . WriteLine ( 'The value of G:' + G ); // Alice will choose the private key a a = 4 ; // a is the chosen private key Console . WriteLine ( 'nThe private key a for Alice:' + a ); x = power ( G a P ); // gets the generated key // Bob will choose the private key b b = 3 ; // b is the chosen private key Console . WriteLine ( 'The private key b for Bob:' + b ); y = power ( G b P ); // gets the generated key // Generating the secret key after the exchange // of keys ka = power ( y a P ); // Secret key for Alice kb = power ( x b P ); // Secret key for Bob Console . WriteLine ( 'nSecret key for the Alice is:' + ka ); Console . WriteLine ( 'Secret key for the Alice is:' + kb ); } } // This code is contributed by Pranay Arora
' ); // A primitive root for P G is taken G = 9 ; document . write ( 'The value of G:' + G + '
' ); // Alice will choose the private key a // a is the chosen private key a = 4 ; document . write ( 'The private key a for Alice:' + a + '
' ); // Gets the generated key x = power ( G a P ); // Bob will choose the private key b // b is the chosen private key b = 3 ; document . write ( 'The private key b for Bob:' + b + '
' ); // Gets the generated key y = power ( G b P ); // Generating the secret key after the exchange // of keys ka = power ( y a P ); // Secret key for Alice kb = power ( x b P ); // Secret key for Bob document . write ( 'Secret key for the Alice is:' + ka + '
' ); document . write ( 'Secret key for the Bob is:' + kb + '
' ); // This code is contributed by Ankita saini < /script>
AusgabeThe value of P : 23 The value of G : 9 The private key a for Alice : 4 The private key b for Bob : 3 Secret key for the Alice is : 9 Secret key for the Bob is : 9
