AES - SBOX雪崩效應問題

我正在製作一個程序，用於驗證 AES 分組密碼的 SBOX 的雪崩效應一定次數的迭代。該程序在輸入列表（狀態）的一個元素中隨機更改一位，然後將 subBytes 操作應用於原始列表和修改後的列表。之後，它計算這兩個結果列表之間不同位的數量。

sbox = [
   0x63, 0x7c, 0x77, 0x7b, 0xf2, 0x6b, 0x6f, 0xc5, 0x30, 0x01,   0x67, 0x2b, 0xfe, 0xd7, 0xab, 0x76,
   0xca, 0x82, 0xc9, 0x7d, 0xfa, 0x59, 0x47, 0xf0, 0xad, 0xd4, 0xa2, 0xaf, 0x9c, 0xa4, 0x72, 0xc0,
   0xb7, 0xfd, 0x93, 0x26, 0x36, 0x3f, 0xf7, 0xcc, 0x34, 0xa5, 0xe5, 0xf1, 0x71, 0xd8, 0x31, 0x15,
   0x04, 0xc7, 0x23, 0xc3, 0x18, 0x96, 0x05, 0x9a, 0x07, 0x12, 0x80, 0xe2, 0xeb, 0x27, 0xb2, 0x75,
   0x09, 0x83, 0x2c, 0x1a, 0x1b, 0x6e, 0x5a, 0xa0, 0x52, 0x3b, 0xd6, 0xb3, 0x29, 0xe3, 0x2f, 0x84,
   0x53, 0xd1, 0x00, 0xed, 0x20, 0xfc, 0xb1, 0x5b, 0x6a, 0xcb, 0xbe, 0x39, 0x4a, 0x4c, 0x58, 0xcf,
   0xd0, 0xef, 0xaa, 0xfb, 0x43, 0x4d, 0x33, 0x85, 0x45, 0xf9, 0x02, 0x7f, 0x50, 0x3c, 0x9f, 0xa8,
   0x51, 0xa3, 0x40, 0x8f, 0x92, 0x9d, 0x38, 0xf5, 0xbc, 0xb6, 0xda, 0x21, 0x10, 0xff, 0xf3, 0xd2,
   0xcd, 0x0c, 0x13, 0xec, 0x5f, 0x97, 0x44, 0x17, 0xc4, 0xa7, 0x7e, 0x3d, 0x64, 0x5d, 0x19, 0x73,
   0x60, 0x81, 0x4f, 0xdc, 0x22, 0x2a, 0x90, 0x88, 0x46, 0xee, 0xb8, 0x14, 0xde, 0x5e, 0x0b, 0xdb,
   0xe0, 0x32, 0x3a, 0x0a, 0x49, 0x06, 0x24, 0x5c, 0xc2, 0xd3, 0xac, 0x62, 0x91, 0x95, 0xe4, 0x79,
   0xe7, 0xc8, 0x37, 0x6d, 0x8d, 0xd5, 0x4e, 0xa9, 0x6c, 0x56, 0xf4, 0xea, 0x65, 0x7a, 0xae, 0x08,
   0xba, 0x78, 0x25, 0x2e, 0x1c, 0xa6, 0xb4, 0xc6, 0xe8, 0xdd, 0x74, 0x1f, 0x4b, 0xbd, 0x8b, 0x8a,
   0x70, 0x3e, 0xb5, 0x66, 0x48, 0x03, 0xf6, 0x0e, 0x61, 0x35, 0x57, 0xb9, 0x86, 0xc1, 0x1d, 0x9e,
   0xe1, 0xf8, 0x98, 0x11, 0x69, 0xd9, 0x8e, 0x94, 0x9b, 0x1e, 0x87, 0xe9, 0xce, 0x55, 0x28, 0xdf,
   0x8c, 0xa1, 0x89, 0x0d, 0xbf, 0xe6, 0x42, 0x68, 0x41, 0x99, 0x2d, 0x0f, 0xb0, 0x54, 0xbb, 0x16
]
sboxInv = [
   0x52, 0x09, 0x6a, 0xd5, 0x30, 0x36, 0xa5, 0x38, 0xbf, 0x40, 0xa3, 0x9e, 0x81, 0xf3, 0xd7, 0xfb,
   0x7c, 0xe3, 0x39, 0x82, 0x9b, 0x2f, 0xff, 0x87, 0x34, 0x8e, 0x43, 0x44, 0xc4, 0xde, 0xe9, 0xcb,
   0x54, 0x7b, 0x94, 0x32, 0xa6, 0xc2, 0x23, 0x3d, 0xee, 0x4c, 0x95, 0x0b, 0x42, 0xfa, 0xc3, 0x4e,
   0x08, 0x2e, 0xa1, 0x66, 0x28, 0xd9, 0x24, 0xb2, 0x76, 0x5b, 0xa2, 0x49, 0x6d, 0x8b, 0xd1, 0x25,
   0x72, 0xf8, 0xf6, 0x64, 0x86, 0x68, 0x98, 0x16, 0xd4, 0xa4, 0x5c, 0xcc, 0x5d, 0x65, 0xb6, 0x92,
   0x6c, 0x70, 0x48, 0x50, 0xfd, 0xed, 0xb9, 0xda, 0x5e, 0x15, 0x46, 0x57, 0xa7, 0x8d, 0x9d, 0x84,
   0x90, 0xd8, 0xab, 0x00, 0x8c, 0xbc, 0xd3, 0x0a, 0xf7, 0xe4, 0x58, 0x05, 0xb8, 0xb3, 0x45, 0x06,
   0xd0, 0x2c, 0x1e, 0x8f, 0xca, 0x3f, 0x0f, 0x02, 0xc1, 0xaf, 0xbd, 0x03, 0x01, 0x13, 0x8a, 0x6b,
   0x3a, 0x91, 0x11, 0x41, 0x4f, 0x67, 0xdc, 0xea, 0x97, 0xf2, 0xcf, 0xce, 0xf0, 0xb4, 0xe6, 0x73,
   0x96, 0xac, 0x74, 0x22, 0xe7, 0xad, 0x35, 0x85, 0xe2, 0xf9, 0x37, 0xe8, 0x1c, 0x75, 0xdf, 0x6e,
   0x47, 0xf1, 0x1a, 0x71, 0x1d, 0x29, 0xc5, 0x89, 0x6f, 0xb7, 0x62, 0x0e, 0xaa, 0x18, 0xbe, 0x1b, 
   0xfc, 0x56, 0x3e, 0x4b, 0xc6, 0xd2, 0x79, 0x20, 0x9a, 0xdb, 0xc0, 0xfe, 0x78, 0xcd, 0x5a, 0xf4, 
   0x1f, 0xdd, 0xa8, 0x33, 0x88, 0x07, 0xc7, 0x31, 0xb1, 0x12, 0x10, 0x59, 0x27, 0x80, 0xec, 0x5f, 
   0x60, 0x51, 0x7f, 0xa9, 0x19, 0xb5, 0x4a, 0x0d, 0x2d, 0xe5, 0x7a, 0x9f, 0x93, 0xc9, 0x9c, 0xef, 
   0xa0, 0xe0, 0x3b, 0x4d, 0xae, 0x2a, 0xf5, 0xb0, 0xc8, 0xeb, 0xbb, 0x3c, 0x83, 0x53, 0x99, 0x61, 
   0x17, 0x2b, 0x04, 0x7e, 0xba, 0x77, 0xd6, 0x26, 0xe1, 0x69, 0x14, 0x63, 0x55, 0x21, 0x0c, 0x7d
]
def subBytes(state):
   result = [None] * 16
   for i in range(len(state)):
       result[i] = sbox[state[i]]
   return result

def subBytesInv(state):
   result = [None] * 16
   for i in range(len(state)):
       result[i] = sboxInv[state[i]]
   return result

#Changes a random bit in one of the elements of the state
def changeBit(state):
   result = state[:]
   result[random.randrange(0,16)] ^= 1 << random.randrange(0,8)
   return result

#Return index of first element that is different in the list
def findDiff(list1, list2):
   for i in range(len(list1)):
       if list1[i] != list2[i]:
           return i
   return -1

#Count number of different bits between two integers
def countDiffBits(a, b):
   return bin(a ^ b).count('1')


#Default array
state=[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]


average = []
for i in range(10000):
   #Create new state with random bit changed
   changed_state = changeBit(state)

   #Apply bin() to each element and print
   print(state)
   sbox_state = subBytes(state) 
   print(list(map(bin,sbox_state)))

   print(changed_state)
   sbox_changed_state = subBytes(changed_state) 
   print(list(map(bin,sbox_changed_state)))

   index = findDiff(state,changed_state)
   print("Changed bits: " + str(countDiffBits(sbox_state[index],sbox_changed_state[index])))

   average.append(countDiffBits(sbox_state[index],sbox_changed_state[index]))

print("Statistics: " + str(Counter(average)))

這是輸出：

Statistics: Counter({5: 2611, 3: 2323, 4: 2236, 2: 1682, 6: 656, 1: 340, 8: 77, 7: 75})

但是，如您所見，對於 10000 次迭代，有 340 次情況在兩個列表之間僅更改了一位，這意味著雪崩效應沒有發生。我現在的問題是，這是否是程式碼中的某種錯誤，或者我是否誤解了這個概念。

然而，如您所見，對於 10000 次迭代，有 340 種情況在兩個列表之間僅更改了一位

這比預期的要高一點。對於 AES SBox，有 24 對輸入（將 02 和 0a 算作與 0a 和 02 相同的一對）具有單個位輸入差分，導致單個位輸出差分；隨機抽取 10000 對，預計會看到 234 對左右。

更新：我看到了問題：您的程式碼將第一組對設置為 0 到 15（不是 255）之間的隨機值。事實證明，AES 有三個這樣的對，它們產生一位差分輸出（02,0a,04,14和08,28）；這給出了預期的 312 個匹配項（在 10,000 個中）；它與您看到的 340 之間的差異可以歸因於隨機性。

我誤解了這個概念。

實際上，AES 並不依賴於 sbox 來雪崩（MixCollumns 和 ShiftRows 的組合很好地做到了這一點）；相反，它依賴於 sbox 對線性和微分特性具有低機率（並且少數 1 位 -> 1 位差分對的存在與此不衝突）

引用自：https://crypto.stackexchange.com/questions/102984