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# 5.6: The ElGamal Cryptosystem


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As mentioned in the previous section, exponentiation in mod $$p$$, where $$p$$ is a prime known to the public, is a good candidate one-way function: It is fast (feasible) in the forward direction; its inverse, being discrete log with respect to a primitive root $$r$$ mod $$p$$ is thought to be infeasible – even when that root is known to the public. This was behind the security of DHKE, and now we discuss how to use this one-way function to set up a more usual public-key cryptosystem: the ElGamal Cryptosystem.

The RSA public-key cryptosystem did its actual encryption by exponentiation in mod $$n=pq$$. The decryption then was by exponentiation with the multiplicative inverse in mod $$\phi(n)$$ of the encryption exponent. The owner of the private key – $$(p,\ q)$$ – knows also the decryption exponent, entirely because $$\phi(n)$$ can be computed.

Similarly, ElGamal uses an elementary arithmetic operation – multiplication of a numerical form of the message by a random number in some mod – to do the scrambling needed for encryption. Enough information is also passed along in the ciphertext so that the intended recipient, who knows the value of a certain discrete log, can cancel out this scrambling multiplication. Here are the details:

Definition: ElGamal Cryptosystem

To start, Alice picks a large prime $$p$$, a primitive root $$r$$ mod $$p$$, and a secret value $$\alpha\in\NN$$ satisfying $$2\le\alpha\le p-1$$. She computes the value $$a=r^\alpha$$ and then posts her ElGamal public [encryption] key $$(p,r,a)$$ on her website.

Alice’s ElGamal private [decryption] key is $$(p,r,\alpha)$$. The association of decryption to encryption keys is by $$\Ee:(p,r,\alpha)\mapsto(p,r,r^\alpha)$$.

The message space is $$\Mm=\{m\in\ZZ\mid 2\le m\le p-1\}$$, which we will assume can be interpreted as meaningful messages encoded numerically by some widely known scheme.

Say Bob wishes to send Alice the cleartext $$m\in\Mm$$. For each new such message $$m$$, he generates a random number $$\beta\in\NN$$ such that $$2\le\beta\le p-2$$ and builds the ciphertext for ElGamal encryption as the two pieces $c=e_{(p,r,a)}(m)=(r^\beta\pmod*{p},\ m\cdot a^\beta\pmod*{p})\ .$

When Alice gets the ciphertext $$c=(c_1,c_2)$$, she can recover the cleartext by ElGamal decryption $d_{(p,r,\alpha)}(c_1,c_2) = c_2\cdot c_1^{p-1-\alpha}\ .$

All of the above parts together form the ElGamal cryptosystem.

We first need to know this is correct, in the sense that

Proposition $$\PageIndex{1}$$

With the notation as above in the Definition of ElGamal Cryptosystem, we have $d_{(p,r,\alpha)}(e_{(p,r,a)}(m)) = m\quad\forall m\in\Mm\ .$

Proof

Just compute: \begin{aligned} d_{(p,r,\alpha)}(e_{(p,r,a)}(m)) &\equiv m\cdot a^\beta\pmod*{p}\cdot(r^\beta)^{p-1-\alpha}\pmod{p}\\ &\equiv m\cdot (r^\alpha)^\beta\cdot(r^\beta)^{p-1-\alpha}\pmod{p}\\ &\equiv m\cdot r^{\alpha\beta+\beta(p-1)-\alpha\beta}\pmod{p}\\ &\equiv m\cdot (r^{p-1})^\beta\pmod{p}\\ &\equiv m\cdot 1^\beta\pmod{p}\\ &\equiv m\pmod{p}\end{aligned} Note that the power $$p-1-\alpha$$ as the exponent of the $$c_1$$ term in decryption is to make $$(c_1^{-1})^\alpha$$ without using negative powers, by applying Theorem 5.1.2.

Graphically:

ElGamal Cryptosystem:

 Alice on public network Bob pick a prime $$p$$ find a primitive root $$r$$ choose $$\alpha\ \mid\ 2\le\alpha\le p-1$$ compute $$a=r^\alpha\pmod{p}$$ publish public key $$\rightarrowtail(p,r,a)$$ download public key given message $$m\in\Mm$$ (so $$2\le m\le p-1$$) choose $$\beta\ \mid\ 2\le\beta\le p-2$$ compute $$c_1=r^\beta\pmod*{p}$$ and $$c_2=m\cdot a^\beta\pmod*{p}$$ receive ciphertext $$(c_1,c_2)\leftarrowtail$$ transmit ciphertext compute cleartext $$m = c_2\cdot c_1^{p-1-\alpha}$$

There is a nice digital signature algorithm associated with ElGamal:

Definition: ElGamal Signature

Suppose Alice has ElGamal private key $$(p,r,\alpha)$$ and wishes to digitally sign the message $$m\in\Mm$$. She first chooses a random $$\gamma\in\NN$$ satisfying $$1<\gamma<p-1$$ and $$\gcd(\gamma,p-1)=1$$.

The digital signature on $$m$$ is $$(r^\gamma\pmod*{p},\gamma^{-1}\cdot(m-\alpha r^\gamma)\pmod*{p-1})$$, where the inverse is taken in mod $$p-1$$.

To verify the signature $$(x,y)$$ on message $$m$$ using Alice’s public key $$(p,r,a)$$, Bob checks to see if $$a^x x^y\equiv r^m\pmod{p}$$: if so, he accepts; if not, he rejects.

Again, we would like to know this does the right thing:

Proposition $$\PageIndex{2}$$

Using the notation as above, Bob will accept all signed messages produced by Alice.

Proof

Assuming the signed message $$(m,x,y)$$ was produced by Alice as above, we compute: \begin{aligned} a^x x^y &\equiv a^{r^\gamma} (r^\gamma)^{\gamma^{-1}(m-\alpha r^\gamma)}\pmod{p}\\ &\equiv (r^\alpha)^{r^\gamma} (r^\gamma)^{\gamma^{-1}(m-\alpha r^\gamma)}\pmod{p}\\ &\equiv r^{\alpha r^\gamma+\gamma\gamma^{-1}(m-\alpha r^\gamma)}\pmod{p}\\ &\equiv r^{\alpha r^\gamma+m-\alpha r^\gamma}\pmod{p}\\ &\equiv r^m\pmod{p}\end{aligned} So Bob will accept.

Graphically:

ElGamal Digital Signatures:

 Alice on public network Bob find prime $$p$$ and primitive root $$r$$ choose $$\alpha\ \mid\ 2\le\alpha\le p-1$$ compute $$a=r^\alpha\pmod{p}$$ publish public key $$\rightarrowtail(p,r,a)$$ download public key given message $$m\in\Mm$$ (so $$2\le m\le p-1$$) pick random $$\gamma$$ s.t. $$1<\gamma Exercise \(\PageIndex{1}$$

1. You instructor still likes the prime $$p=11717$$ with primitive root $$r=103$$ from an earlier exercise (Exercise 5.5.1(2)) on DHKE. In addition, your instructor has calculated the value $$a=1020$$ to complete an ElGamal public key $$(p,r,a)=(11717,103,1020)$$.

Using this public key, you want to send a message to your instructor, which should consist of the number $$42$$ (it is, after all, the answer to “life, the universe, and everything”). What ciphertext will you send? Show your work!

2. Now your instructor wants to send you your grade on a recent test by e-mail, and to prove that this e-mail does in fact originate with your instructor, the email contains both the score value of $$97$$ and the addendum “This score value signed with an ElGamal Digital signature using my public key [the same instructor’s public key as above in Exercise 5.6.1(1) being $$(p,r,a)=(11717,103,1020)$$]; the signature has the value $$(6220, 10407)$$.”

Do you accept this as truly coming from your instructor? Show your work!

3. Create an ElGamal public key and e-mail it to your instructor. Wait for a reply message which is ElGamal encrypted, then mail the cleartext back to your instructor.

Also, use your public key to sign the number (=message) $$17$$. Send the signed number to your instructor and wait to hear if the signature is accepted or not.

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