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When reading up on the Rijndael key schedule, I learned that the master key itself is used as a round key directly before the key schedule generates additional round keys. When a 256-bit key is used, the first half of the key goes to the first AddRoundKey and the second half goes to the second. This means that, if the 256-bit master key is composed of a single 128-bit value repeated twice, the first and second round key will be the same. I assume the cipher is not designed to remain secure under identical round keys, or there wouldn't be the need to use a key schedule.

Example key schedule test vectors with a 256-bit null key show round keys 1 and 2 are identical:

 key: 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 
      00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 

rk01: 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 
rk02: 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 00 
rk03: 62 63 63 63 62 63 63 63 62 63 63 63 62 63 63 63 
rk04: aa fb fb fb aa fb fb fb aa fb fb fb aa fb fb fb 
rk05: 6f 6c 6c cf 0d 0f 0f ac 6f 6c 6c cf 0d 0f 0f ac 
rk06: 7d 8d 8d 6a d7 76 76 91 7d 8d 8d 6a d7 76 76 91 
rk07: 53 54 ed c1 5e 5b e2 6d 31 37 8e a2 3c 38 81 0e 
rk08: 96 8a 81 c1 41 fc f7 50 3c 71 7a 3a eb 07 0c ab 
rk09: 9e aa 8f 28 c0 f1 6d 45 f1 c6 e3 e7 cd fe 62 e9 
rk10: 2b 31 2b df 6a cd dc 8f 56 bc a6 b5 bd bb aa 1e 
rk11: 64 06 fd 52 a4 f7 90 17 55 31 73 f0 98 cf 11 19 
rk12: 6d bb a9 0b 07 76 75 84 51 ca d3 31 ec 71 79 2f 
rk13: e7 b0 e8 9c 43 47 78 8b 16 76 0b 7b 8e b9 1a 62 
rk14: 74 ed 0b a1 73 9b 7e 25 22 51 ad 14 ce 20 d4 3b 
rk15: 10 f8 0a 17 53 bf 72 9c 45 c9 79 e7 cb 70 63 85 

Do two identical round keys cause any security issues in Rjndael?

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Yes. less data complexity required to retrieve the master key.

Sub-keys can be represented by linear combination. try to xor between 2 consecutive sub-keys, you will find relations.

The paper (Low Data Complexity Attacks on AES) shows in observation 3 (page 8) a relation in consecutive sub-keys that can be exploited.

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