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문제

여기 암호화해서 전송하고싶은 데이터가 32bit unsigned int 8개 형태로 주어져있다.

N1 N2 N3 ... N8

우선 이 데이터가 올바른 데이터인지 확인해주는 checksum을 구해서 9번째 데이터로 추가해줬다. (checksum은 데이터의 무결성을 확인하는데 자주 사용되는 방식이다.)

N9 = sigma(Ni) mod 2^32

그다음 특정 32bit 키값 K를 이용해서 이 9개의 숫자를 암호화하려고 한다. 암호화는 XOR 연산을 통해 이루어진다고 보자. (XOR은 데이터의 암호화를 위해 자주 사용되는 연산자다.)

M1 = N1 ^K, M2 = N2 ^ K ... M9 = N9 ^ K

이렇게 해서 만들어진 M1 ~ M9 9개의 데이터를 전송하면 K값을 알고있는 사람은 원래 데이터 N1 ~ N8을 알아낼 수 있다. 그리고 K값을 모르는 사람은 원래 데이터 N1 ~ N8을 알아낼 수.....있다!

놀랍게도 위의 방식에는 취약점이 존재한다. 취약점을 발견해서 9개의 정수 M1 ~ M9 가 16진수 형태로 주어졌을 때, 키값 K를 구하는 프로그램을 작성해보자.

입력

입력의 첫째 줄에는 테스트데이터의 개수 T가 주어진다. T는 1000을 넘지 않는다.

그 다음 각 테스트데이터마다 9개의 정수 M1 ~ M9 가 16진수 형태로 주어진다. 16진수는 '0' ~ '9' 그리고 소문자로만 이루어져 있고, 불필요한 0은 들어오지 않는다. 9개의 숫자가 한 줄에 모두 입력되지 않을 수도 있다.

출력

각 테스트데이터마다 한줄에 하나씩 키값 K를 소문자 16진수 형태로 출력한다. 불필요한 0은 출력하지 않는다.

예제 입력 1

8
1 1 1 1 1 1 1 1 8
3 2 3 2 3 2 3 2 6
3 4 4 7 7 b a 2 2e
e1 13 ce 28 ca 6 ab 46 a6d
b08 49e2 6128 f27 8cf2 bc50 7380 7fe1 723b
4eba eb4 a352 fd14 6ac1 eed1 dd06 bb83 392bc
ef593c08 847e522f 74c02b9c 26f3a4e1 e2720a01 6fe66007
7a4e96ad 6ee5cef6 3853cd88
60202fb8 757d6d66 9c3a9525 fbcd7983 82b9571c ddc54bab 853e52da
22047c88 e5524401

예제 출력 1

0
2
6
1c6
4924afc7
ffff95c5
546991d
901c4a16

힌트

원문으로 볼 시 스포일러 주의

[{"problem_id":"3925","problem_lang":"0","title":"\uc554\ud638\ud654\uc758 \ucde8\uc57d\uc810 \ucc3e\uae30","description":"<p>\uc5ec\uae30 \uc554\ud638\ud654\ud574\uc11c \uc804\uc1a1\ud558\uace0\uc2f6\uc740 \ub370\uc774\ud130\uac00 32bit unsigned int 8\uac1c \ud615\ud0dc\ub85c \uc8fc\uc5b4\uc838\uc788\ub2e4.<\/p>\r\n\r\n<p>N1 N2 N3 ... N8<\/p>\r\n\r\n<p>\uc6b0\uc120 \uc774 \ub370\uc774\ud130\uac00 \uc62c\ubc14\ub978 \ub370\uc774\ud130\uc778\uc9c0 \ud655\uc778\ud574\uc8fc\ub294 checksum\uc744 \uad6c\ud574\uc11c 9\ubc88\uc9f8 \ub370\uc774\ud130\ub85c \ucd94\uac00\ud574\uc92c\ub2e4. (checksum\uc740 \ub370\uc774\ud130\uc758 \ubb34\uacb0\uc131\uc744 \ud655\uc778\ud558\ub294\ub370 \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \ubc29\uc2dd\uc774\ub2e4.)<\/p>\r\n\r\n<p>N9 = sigma(Ni) mod 2^32<\/p>\r\n\r\n<p>\uadf8\ub2e4\uc74c \ud2b9\uc815 32bit \ud0a4\uac12 K\ub97c \uc774\uc6a9\ud574\uc11c \uc774 9\uac1c\uc758 \uc22b\uc790\ub97c \uc554\ud638\ud654\ud558\ub824\uace0 \ud55c\ub2e4. \uc554\ud638\ud654\ub294 XOR \uc5f0\uc0b0\uc744 \ud1b5\ud574 \uc774\ub8e8\uc5b4\uc9c4\ub2e4\uace0 \ubcf4\uc790. (XOR\uc740 \ub370\uc774\ud130\uc758 \uc554\ud638\ud654\ub97c \uc704\ud574 \uc790\uc8fc \uc0ac\uc6a9\ub418\ub294 \uc5f0\uc0b0\uc790\ub2e4.)<\/p>\r\n\r\n<p>M1 = N1 ^K, M2 = N2 ^ K ... M9 = N9 ^ K<\/p>\r\n\r\n<p>\uc774\ub807\uac8c \ud574\uc11c \ub9cc\ub4e4\uc5b4\uc9c4 M1 ~ M9 9\uac1c\uc758 \ub370\uc774\ud130\ub97c \uc804\uc1a1\ud558\uba74 K\uac12\uc744 \uc54c\uace0\uc788\ub294 \uc0ac\ub78c\uc740 \uc6d0\ub798 \ub370\uc774\ud130 N1 ~ N8\uc744 \uc54c\uc544\ub0bc \uc218 \uc788\ub2e4. \uadf8\ub9ac\uace0 K\uac12\uc744 \ubaa8\ub974\ub294 \uc0ac\ub78c\uc740 \uc6d0\ub798 \ub370\uc774\ud130 N1 ~ N8\uc744 \uc54c\uc544\ub0bc \uc218.....\uc788\ub2e4!<\/p>\r\n\r\n<p>\ub180\ub78d\uac8c\ub3c4 \uc704\uc758 \ubc29\uc2dd\uc5d0\ub294 \ucde8\uc57d\uc810\uc774 \uc874\uc7ac\ud55c\ub2e4. \ucde8\uc57d\uc810\uc744 \ubc1c\uacac\ud574\uc11c 9\uac1c\uc758 \uc815\uc218 M1 ~ M9 \uac00 16\uc9c4\uc218 \ud615\ud0dc\ub85c \uc8fc\uc5b4\uc84c\uc744 \ub54c, \ud0a4\uac12 K\ub97c \uad6c\ud558\ub294 \ud504\ub85c\uadf8\ub7a8\uc744 \uc791\uc131\ud574\ubcf4\uc790.<\/p>\r\n","input":"<p>\uc785\ub825\uc758 \uccab\uc9f8 \uc904\uc5d0\ub294 \ud14c\uc2a4\ud2b8\ub370\uc774\ud130\uc758 \uac1c\uc218 T\uac00 \uc8fc\uc5b4\uc9c4\ub2e4. T\ub294 1000\uc744 \ub118\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n\r\n<p>\uadf8 \ub2e4\uc74c \uac01 \ud14c\uc2a4\ud2b8\ub370\uc774\ud130\ub9c8\ub2e4 9\uac1c\uc758 \uc815\uc218 M1 ~ M9 \uac00 16\uc9c4\uc218 \ud615\ud0dc\ub85c \uc8fc\uc5b4\uc9c4\ub2e4. 16\uc9c4\uc218\ub294 &#39;0&#39; ~ &#39;9&#39; \uadf8\ub9ac\uace0 \uc18c\ubb38\uc790\ub85c\ub9cc \uc774\ub8e8\uc5b4\uc838 \uc788\uace0, \ubd88\ud544\uc694\ud55c 0\uc740 \ub4e4\uc5b4\uc624\uc9c0 \uc54a\ub294\ub2e4. 9\uac1c\uc758 \uc22b\uc790\uac00 \ud55c \uc904\uc5d0 \ubaa8\ub450 \uc785\ub825\ub418\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4.<\/p>\r\n","output":"<p>\uac01 \ud14c\uc2a4\ud2b8\ub370\uc774\ud130\ub9c8\ub2e4 \ud55c\uc904\uc5d0 \ud558\ub098\uc529 \ud0a4\uac12 K\ub97c \uc18c\ubb38\uc790 16\uc9c4\uc218 \ud615\ud0dc\ub85c \ucd9c\ub825\ud55c\ub2e4. \ubd88\ud544\uc694\ud55c 0\uc740 \ucd9c\ub825\ud558\uc9c0 \uc54a\ub294\ub2e4.<\/p>\r\n","hint":"<p>\uc6d0\ubb38\uc73c\ub85c \ubcfc \uc2dc \uc2a4\ud3ec\uc77c\ub7ec \uc8fc\uc758<\/p>\r\n","original":"0","problem_lang_code":"\ud55c\uad6d\uc5b4"},{"problem_id":"3925","problem_lang":"1","title":"Leaky Cryptography","description":"<p>The ACM ICPC judges are very careful about not leaking their problems, and all communications are encrypted. However, one does sometimes make mistakes, like using too weak an encryption scheme. Here is an example of that.<\/p>\r\n\r\n<p>The encryption chosen was very simple: encrypt each chunk of the input by flipping some bits according to a shared key. To provide reasonable security, the size of both chunk and key is 32 bits.<\/p>\r\n\r\n<p>That is, suppose the input was a sequence of \\(m\\) 32-bit integers.<\/p>\r\n\r\n<p>\\[N_1 N_2 N_3 \\dots N_m\\]<\/p>\r\n\r\n<p>After encoding with the key \\(K\\) it becomes the following sequence of \\(m\\) 32-bit integers.<\/p>\r\n\r\n<p>\\[(N_1 \\wedge &nbsp;K) (N_2 \\wedge &nbsp;K) (N_3 \\wedge &nbsp;K) \\dots (N_m \\wedge &nbsp;K)\\]<\/p>\r\n\r\n<p>where \\((a \\wedge b)\\) is the bitwise exclusive or of a and b.<\/p>\r\n\r\n<p>Exclusive or is the logical operator which is 1 when only one of its operands is 1, and 0 otherwise. Here is its definition for 1-bit integers.<\/p>\r\n\r\n<p>\\[0 \\oplus &nbsp;0 = 0 ~~~ 0 \\oplus &nbsp; 1 = 1 &nbsp; \\\\ 1 \\oplus &nbsp; 0 = 1 ~~~ 1 \\oplus &nbsp; 1 = 0\\]<\/p>\r\n\r\n<p>As you can see, it is identical to addition modulo 2. For two 32-bit integers \\(a\\) and \\(b\\), their bitwise exclusive or \\(a \\wedge b\\) is defined as follows, using their binary representations, composed of 0&rsquo;s and 1&rsquo;s.<\/p>\r\n\r\n<p>\\[a \\wedge b = a_{31} \\dots a_1a_0 \\wedge b_{31} \\dots b_1b_0 = c_{31} \\dots c_1c_0\\]<\/p>\r\n\r\n<p>where<\/p>\r\n\r\n<p>\\[c_i = a_i \\oplus b_i (i=0,1,\\dots31)\\]<\/p>\r\n\r\n<p>For instance, using binary notation, 11010110 &and; 01010101 = 10100011, or using hexadecimal, d6 &and; 55 = a3.<\/p>\r\n\r\n<p>Since this kind of encryption is notoriously weak to statistical attacks, the message has to be compressed in advance, so that it has no statistical regularity. We suppose that \\(N_1 N_2 \\dots N_m\\) is already in compressed form.<\/p>\r\n\r\n<p>However, the trouble is that the compression algorithm itself introduces some form of regularity: after every 8 integers of compressed data, it inserts a checksum, the sum of these integers. That is, in the above input, \\(N_9 = \\sum_{i=1}^{8}{N_i} = N_1 + \\cdots + N_8\\) , where additions are modulo 2<sup>32<\/sup>.<\/p>\r\n\r\n<p>Luckily, you could intercept a communication between the judges. Maybe it contains a problem for the finals!<\/p>\r\n\r\n<p>As you are very clever, you have certainly seen that you can easily find the lowest bit of the key, denoted by \\(K_0\\). On the one hand, if \\(K_0\\) = 1, then after encoding, the lowest bit of \\(\\sum_{i=1}^{8}{ N_i \\wedge K}\\) is unchanged, as \\(K_0\\) is added an even number of times, but the lowest bit of \\(N_9 \\wedge K\\) is changed, so they shall differ. On the other hand, if \\(K_0\\) = 0, then after encoding, the lowest bit of \\(\\sum_{i=1}^{8}{N_i \\wedge K} \\) shall still be identical to the lowest bit of \\(N_9 \\wedge K\\), as they do not change. For instance, if the lowest bits after encoding are 1 1 1 1 1 1 1 1 1 then \\(K_0\\) must be 1, but if they are 1 1 1 1 1 1 1 0 1 then \\(K_0\\) must be 0.<\/p>\r\n\r\n<p>So far, so good. Can you do better?<\/p>\r\n\r\n<p>You should find the key used for encoding.<\/p>\r\n","input":"<p>The input starts with a line containing only a positive integer S, indicating the number of datasets in the input. S is no more than 1000.<\/p>\r\n\r\n<p>It is followed by S datasets. Each dataset is composed of nine 32-bit integers corresponding to the first nine chunks of a communication. They are written in hexadecimal notation, using digits &lsquo;0&rsquo; to &lsquo;9&rsquo; and lowercase letters &lsquo;a&rsquo; to &lsquo;f&rsquo;, and with no leading zeros. They are separated by a space or a newline. Each dataset is ended by a newline.<\/p>\r\n","output":"<p>For each dataset you should output the key used for encoding. Each key shall appear alone on its line, and be written in hexadecimal notation, using digits &lsquo;0&rsquo; to &lsquo;9&rsquo; and lowercase letters &lsquo;a&rsquo; to &lsquo;f&rsquo;, and with no leading zeros.<\/p>\r\n","hint":"","original":"1","problem_lang_code":"\uc601\uc5b4"}]