![]() The procedure then becomes: create a random integer $u$ modulo $p$ and compute $g = u^ \mod p$. Nevertheless, some people get nervous in the face of probabilities, and will feel safer if we test the generator. so small that you will not hit it in practice (the overall security of Diffie-Hellman relies on the practical impossibility of obtaining events which are billions of times more probable than that). Then, to get the "generator" $g$, we can just use any random integer modulo $p$: the probability that $q$ is not a divisor of the order of a random non-zero integer modulo $p$ if $1/q$, i.e. Note that producing a random prime $p$ would already yield a "large enough" $q$ with overwhelming probability (but we wound not know the value of $q$, only that it is very improbable that the largest prime factor of $p-1$ is smaller than $2t$ bits). This is what is done for generating DSA key parameters, which have the same structure than Diffie-Hellman key parameters (see the DSA standard). Namely, we create a random prime $q$ of size $2t$ bits, then we produce $p = qr 1$ for random values of $r$ until we hit a prime. We normally want to know $q$, so $q$ is generated first, then $p$. Since $n$ necessarily divides $p-1$, $q$ divides $p-1$. "Large enough" means "of length at least $2t$ bits if we target $t$-bit security". Bob and Alice won’t notice any problem and may assume their communication is encrypted, but in reality, Malory can decrypt, read, modify, and then re-encrypt all their conversations.In the general case, for proper security with Diffie-Hellman, we need a value $g$ such that the order of $g$ (the smallest integer $n \geq 1$ such that $g^n = 1 \mod p$) is a multiple of a large enough prime $q$. Step 5: If Alice uses S 1 as a key to encrypt a later message to Bob, Malory can decrypt it, re-encrypt it using S 2, and send it to Bob. Malory intercepts Alice’s public value (g a(mod p)), block it from reaching Bob, and instead sends Bob her own public value (g c(modp)) and Malory intercepts Bob’s public value (g b(mod p)), block it from reaching Alice, and instead sends Alice her own public value (g d (modp))Īlice will compute a key S 1=g da(mod p), and Bob will compute a different key, S 2=g cb(mod p) Let Alice pick a private random number a and let Bob pick a private random number b, Malory picks 2 random numbers c and d. Step 1: Selected public numbers p and g, p is a prime number, called the “modulus” and g is called the base. Step by Step explanation of this process: Man in the Middle (MITM) against Diffie-Hellman:Ī malicious Malory, that has a MitM (man in the middle) position, can manipulate the communications between Alice and Bob, and break the security of the key exchange. If p and g have thousands of bits, then the best-known algorithms to compute discrete logs, although faster than plain brute force, will still take millions of years to compute.Įven with its immunity to brute force, it’s vulnerable to MITM (man in the middle position). With all her knowledge, she still can’t compute the secret key S, as it turns out, if p and g are properly chosen, it’s very, very hard for her to do.įor instance, you could brute force it and try all the options, but The calculations (mod p) make the discrete log calculation super slow when the numbers are large. #DIFFIE HELLMAN CALCULATOR MOD#Let’s assume that the eavesdropper EVE knows the public values p and g like everyone else, and from her eavesdropping, she learns the values exchanged by Alice and Bob, gᵃ mod p and gᵇ mod p, as well. This process is done using some simple algebra, prime numbers, and properties of modular arithmetic. Protocol: To generate new secret keys, run a two-message key exchange protocol.One-time setup: We define some public parameters that are used by everyone forever. ![]() The key exchange procedure has two steps :
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