Update content of files

data/core.telegram.org/mtproto/auth_key.html

@@ -99,7 +99,7 @@
 <li>encrypted_answer := AES256_ige_encrypt (answer_with_hash, tmp_aes_key, tmp_aes_iv); here, tmp_aes_key is a 256-bit key, and  tmp_aes_iv is a 256-bit initialization vector. The same as in all the other instances that use AES encryption, the encrypted data is padded with random bytes to a length divisible by 16 immediately prior to encryption.</li>
 </ul>
 <p>Following this step, new_nonce is still known to client and server only. The client is certain that it is the server that responded and that the response was generated specifically in response to client query req_DH_params, since the response data are encrypted using new_nonce.</p>
-<p>Client is expected to check whether <strong>p = dh_prime</strong> is a safe 2048-bit prime (meaning that both <strong>p</strong> and <strong>(p-1)/2</strong> are prime, and that 2<sup>2047 &lt; p &lt; 2</sup>2048), and that <strong>g</strong> generates a cyclic subgroup of prime order <strong>(p-1)/2</strong>, i.e. is a quadratic residue <strong>mod p</strong>. Since <strong>g</strong> is always equal to 2, 3, 4, 5, 6 or 7, this is easily done using quadratic reciprocity law, yielding a simple condition on <strong>p mod 4g</strong> — namely, <strong>p mod 8 = 7</strong> for <strong>g = 2</strong>; <strong>p mod 3 = 2</strong> for <strong>g = 3</strong>; no extra condition for <strong>g = 4</strong>; <strong>p mod 5 = 1 or 4</strong> for <strong>g = 5</strong>; <strong>p mod 24 = 19 or 23</strong> for <strong>g = 6</strong>; and <strong>p mod 7 = 3, 5 or 6</strong> for <strong>g = 7</strong>. After <strong>g</strong> and <strong>p</strong> have been checked by the client, it makes sense to cache the result, so as not to repeat lengthy computations in future.</p>
+<p>Client is expected to check whether <strong>p = dh_prime</strong> is a safe 2048-bit prime (meaning that both <strong>p</strong> and <strong>(p-1)/2</strong> are prime, and that 2^2047 &lt; p &lt; 2^2048), and that <strong>g</strong> generates a cyclic subgroup of prime order <strong>(p-1)/2</strong>, i.e. is a quadratic residue <strong>mod p</strong>. Since <strong>g</strong> is always equal to 2, 3, 4, 5, 6 or 7, this is easily done using quadratic reciprocity law, yielding a simple condition on <strong>p mod 4g</strong> — namely, <strong>p mod 8 = 7</strong> for <strong>g = 2</strong>; <strong>p mod 3 = 2</strong> for <strong>g = 3</strong>; no extra condition for <strong>g = 4</strong>; <strong>p mod 5 = 1 or 4</strong> for <strong>g = 5</strong>; <strong>p mod 24 = 19 or 23</strong> for <strong>g = 6</strong>; and <strong>p mod 7 = 3, 5 or 6</strong> for <strong>g = 7</strong>. After <strong>g</strong> and <strong>p</strong> have been checked by the client, it makes sense to cache the result, so as not to repeat lengthy computations in future.</p>
 <p>If the verification takes too long time (which is the case for older mobile devices), one might initially run only 15 Miller—Rabin iterations for verifying primeness of <strong>p</strong> and <strong>(p - 1)/2</strong> with error probability not exceeding one billionth, and do more iterations later in the background.</p>
 <p>Another optimization is to embed into the client application code a small table with some known “good” couples <strong>(g,p)</strong> (or just known safe primes <strong>p</strong>, since the condition on <strong>g</strong> is easily verified during execution), checked during code generation phase, so as to avoid doing such verification during runtime altogether. Server changes these values rarely, thus one usually has to put the current value of server&#39;s <strong>dh_prime</strong> into such a table. For example, current value of <strong>dh_prime</strong> equals (in big-endian byte order)</p>
 <pre><code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code></pre>