Back Diffie-Hellman is a standard method of Alice and Bob being able to communicate, and end up with the same secret encryption key. It is used in many applications, and uses two numbers (G and N) for the first part of the calculation (of which N must be a prime number). Recall the mathematics of Diffie-Hellman: Given public. Stack Exchange Network. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Diffie Hellman calculator in Python and my first story. Paper key lists transported by a trusted courier. The Diffie - Hellman key exchange method allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure channel. This key can then be used to encrypt subsequent communications using. Diffie-Hellman (D-H) is a public key algorithm used for producing a shared secret key. It is described in RFC 2631 and Public Key Cryptography Standard (PKCS) #3. To share a secret key between two parties, both parties calculate the shared secret key using their own private key and the other party's public key. Dirty Diffie-Hellman (Like dirty Santa, but geekier) Crappy PHP script for a simple Diffie-Hellman key exchange calculator. I guess I could have used Javascript instead of PHP, but I.
- Ecc Diffie Hellman Calculator
- Diffie Hellman Merkle Key Exchange Calculator
- Diffie-hellman Key Exchange Calculator
- Diffie-hellman Calculator
Take on the roles of Alice and Bob! Exchange secret keys using the Diffie-Hellman key exchange method!! Use your keys to encrypt messages!!!
The Diffie-Hellman key exchange uses a large prime p and a primitive root g of this prime. These numbers are both public.
To start the key exchange process, Alice chooses a secret number a less than the large prime, and computes ga (mod p). Alice sends this answer, call it A, to Bob. Bob now chooses his own secret number b, and computes gb (mod p). Bob sends this answer, call it B, to Alice.
Finally, Alice computes Ba (mod p), and Bob computes Ab (mod p). They both get the same answer, but no-one else will know this secret answer, because only Alice knows a, and only Bob knows b. This secret answer is their private key, which they can use to encrypt messages.
[You may wonder why someone intercepting Alice and Bob's communication can't solve gx = A (mod p) to calculate Alice's secret number a. This is a hard problem, known as the discrete logarithm problem. That this is difficult is the strength of this method of key exchange.]
First you must be Alice. Choose a large prime from the list below (or one of your own choice) and a corresponding primitive root of that large prime. Then choose a secret number which is smaller than your large prime.
- 22953686867719691230002707821868552601124472329079 primitive root 11
- 30762542250301270692051460539586166927291732754961 primitive root 7
- 29927402397991286489627837734179186385188296382227 primitive root 2
- 95647806479275528135733781266203904794419563064407 primitive root 5
- 48705091355238882778842909230056712140813460157899 primitive root 6
- 53542885039615245271174355315623704334284773568199 primitive root 3
- 622288097498926496141095869268883999563096063592498055290461 primitive root 2
- 610692533270508750441931226384209856405876657993997547171387 primitive root 2
- 4669523849932130508876392554713407521319117239637943224980015676156491 primitive root 3
- 4906275427767802358357703730938087362176142642699093827933107888253709 primitive root 2
- 18532395500947174450709383384936679868383424444311405679463280782405796233163977 primitive root 5
- 282755483533707287054752184321121345766861480697448703443857012153264407439766013042402571 primitive root 2
Diffie-Hellman key agreement protocol uses modular exponentiation and calls for use of special prime numbers. If you ever wondered why, I’ll try to explain.
Diffie-Hellman key agreement
Ecc Diffie Hellman Calculator
The “classical” Diffie-Hellman key exchange also known as Finite Field Diffie-Hellman uses one type of operation — modular exponentiation — and two secrets for two communication peers to arrive at a single shared secret.
The protocol requires a prime number and a number that is a so-called “generator” number to be known to both peers. This is usually achieved by either the server sending those values to the client (e.g. in TLS before v1.3 or in some SSH key exchange types) or by distributing them ahead of time to the peers (e.g. in IPSec and TLS 1.3).
When both peers know which parameters to use, they generate a random number, perform modular exponentiation with this random number, group generator and prime and send the result to the other party as a “key share”. That other party takes this key share and uses it as the generator to perform modular exponentiation again. The result of that second operation is the agreed key share.
If we define as the generator, as the prime, as the server selected random, as the server key share, as the client selected random and as the client key share, and being the agreed upon secret, the server performs following operations:
Client performs following operation:
Because both parties agree on the same .
Unfortunately both peers need to operate on a value provided by the other party (not necessarily trusted or authenticated) and their secret value at the same time. This calls for the the prime number used to have some special properties.
Modular exponentiation
The basic operation we’ll be dealing with is modular exponentiation. The simple way to explain it is that we take a number, raise it to some power. Then we take that result and divide it by a third number. The remainder of that division is our result.
For 2^10 mod 12, the calculation will go as follows, first exponentiation:
Then division:
So the result is 4.
One of the interesting properties of modular exponentiation, is that it is cyclic. If we take a base number and start raising it to higher and higher powers, we will be getting the same numbers in the same order:
This comes from the fact that in modulo arithmetic, for addition, subtraction, multiplication and exponentiation, the order in which the modulo operations are made does not matter; a + b mod c is equal to (a mod c + b mod c) mod c. Thus if we try to calculate the example for 3^17 mod 14 we can write it down as ((3^6 mod 14) * (3^6 mod 14) * (3^5 mod 14)) mod 14. Then the calculation is reduced to 1 * 1 * 3^5 mod 14.
The inverse of modular exponentiation is discrete logarithm, in which for a given base and modulus, we look for exponent that will result in given number:
Where g, m and n are given, we’re looking for x.
Because there are no fast algorithms for calculating discrete logarithm, is one of the reasons we can use modulo exponentiation as the base of Diffie-Hellman algorithm.
Cyclic groups
Let’s see what happens if we start calculating results of modular exponentiation for 14 with different bases:
Neither of the numbers can generate all of the numbers that are smaller than the integer we calculate the modulo operation with. In other words, there is no generator (in number theoretic sense) that generates the whole group.
To find such numbers, we need to start looking at prime numbers.
Cyclic groups modulo prime
Let’s see what happens for 13:
The obvious result is that we now have 4 generators — 2, 6, 7 and 11 generate the whole group.
But there is other result hiding. Let’s see the results for 19, but with sizes of those groups shown:
Note that all the sizes of those groups are factors of 18 – that is p-1.
The third observation we can draw from those results is that, for any number the size of the group of the generated elements will be at most as large as the size of the base number.
With 19, if we take generator 8, the size of its subgroup is 6. But size of subgroups of 7, 18, 11 and 12 is respectively 3, 2, 3 and 6.
Thus, not only is the subgroup much smaller than the full group, it is also impossible to “escape” from it.
Safe primes
We saw that for primes, some bases are better than others (look into finite fields to learn more).
As we noticed, sizes of all groups are factors of the prime less one (see Fermat’s little theorem for proof of this). Of course, with the exception of 2, all primes are odd numbers, so p-1 will always be divisible by two – it will be a composite number. But q = (p-1)/2 doesn’t have to be composite. Indeed, we call primes for which q is also prime safe primes.
Let’s see what happens if we calculate groups of such a prime:
Diffie Hellman Merkle Key Exchange Calculator
The groups look very different to the ones we saw previously, with the exception of bases 0, 1 and p-1, all groups are relatively large – 11 or 22 elements.
Diffie-hellman Key Exchange Calculator
One interesting observation we can make about bases that have group order of 2q, is that even exponents will remain in a group of size 2q while odd will move to a group with order of q. Thus we can say that use of generator that is part of group order of 2q will leak the least significant bit of the exponent.
Diffie-hellman Calculator
That’s why protecting against small subgroup attacks with safe primes is so easy, it requires comparing the peer’s key share against just 3 numbers. It’s also the reason why it’s impossible to “backdoor” the parameters (prime and generator), as every generator is a good generator.