ESM Cipher Moustafa Mahmoud Abd El-Rahem Khalil
[email protected] Cairo, Egypt
Abstract I present a new type of ciphers based on the ESM (The set The Set of Equivalent Swapped Multiples) which is the set of pairs of numbers that their multiple equal their multiple with their digits swapped position like the pair (43, 68) → (43 ∗ 68 = 34 ∗ 86 = 2924) or the pair (36, 42) → (36 ∗ 42 = 63 ∗ 24 = 1512)
1. Introduction 𝑇ℎ𝑒 𝑑𝑖𝑟𝑒𝑐𝑡 𝑙𝑖𝑛𝑘: ℎ𝑡𝑡𝑝𝑠://𝑤𝑤. 𝑎𝑐𝑎𝑑𝑒𝑚𝑖𝑎. 𝑒𝑑𝑢/16131410 /𝑇ℎ𝑒_𝑆𝑒𝑡_𝑜𝑓_𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡_𝑆𝑤𝑎𝑝𝑝𝑒𝑑_𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠_𝐸𝑆𝑀_ 𝐼𝑠 𝑚𝑦 𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙 𝑟𝑒𝑠𝑒𝑎𝑟𝑐ℎ 𝑝𝑎𝑝𝑒𝑟 𝑡ℎ𝑎𝑡 𝑑𝑒𝑓𝑖𝑛𝑒𝑠 𝑡ℎ𝑒 𝐸𝑆𝑀 (𝑇ℎ𝑒 𝑆𝑒𝑡 𝑜𝑓 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑆𝑤𝑎𝑝𝑝𝑒𝑑 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠). 12 13 14 21 23 24 26 28 31 32 34 36 39 41 42 43 46 48 62 63 64 68 69 82 84 86 93 96
{42,63,84} {62,93} {82} {24,36,48} {64,96} {21,63,84} {31,93} {41} {26,39} {46,69} {86} {21,42,84} {31,62} {28} {12,36,48} {68} {32,96} {21,42,63} {13,39} {12,24,48} {23,69} {43} {32,64} {14} {12,24,36} {34} {13,26} {23,46}
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𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑙𝑢𝑚𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑎𝑏𝑙𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑥 𝑤ℎ𝑒𝑟𝑒 𝑥 ∈ {10 ≤ 𝑛 ≤ 99 | 𝑛 ∈ 𝑁}. 𝑇ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑙𝑢𝑚𝑛 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑡ℎ𝑒 𝐸𝑆𝑀 𝑜𝑓 𝑒𝑣𝑒𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑙𝑢𝑚𝑛. 𝑇ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑙𝑢𝑚𝑛 𝑑𝑜𝑒𝑠𝑛′ 𝑡 𝑖𝑛𝑐𝑙𝑢𝑑𝑒 𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠, 𝑙𝑖𝑘𝑒 {11,22, … ,88,99} 𝑜𝑟 𝑎 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡ℎ𝑎𝑡 ℎ𝑎𝑣𝑒 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒, 𝑙𝑖𝑘𝑒 {(15,51), (67,76), … }. 𝑇ℎ𝑒 𝐸𝑆𝑀 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 𝑑𝑜𝑒𝑠𝑛′ 𝑡 𝑖𝑛𝑐𝑙𝑢𝑑𝑒 𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑥 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑙𝑢𝑚𝑛, 𝑙𝑖𝑘𝑒, 𝑡ℎ𝑒 𝐸𝑆𝑀 {42,63,84} 𝑑𝑜𝑒𝑠𝑛′ 𝑡 𝑖𝑛𝑐𝑙𝑢𝑑𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 21 𝑎𝑠 𝑖𝑡 ′ 𝑠 𝑎 𝑡𝑟𝑖𝑣𝑖𝑎𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 12. 𝑇ℎ𝑒 𝑢𝑛𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝐸𝑆𝑀𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑙𝑢𝑚𝑛𝑠 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑡𝑜 𝑡ℎ𝑒 𝑢𝑛𝑖𝑜𝑛 𝑜𝑓 𝑡ℎ𝑒 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑙𝑢𝑚𝑛. 𝑇ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑚𝑎𝑡𝑟𝑖𝑥 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑡ℎ𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑥 𝑤𝑖𝑡ℎ 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑟𝑜𝑤 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 ℎ𝑜𝑤 𝑚𝑎𝑛𝑦 𝑡𝑖𝑚𝑒𝑠 𝑡ℎ𝑒 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑎𝑝𝑝𝑒𝑎𝑟𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑙𝑢𝑚𝑛 𝑖𝑛 𝑡ℎ𝑒 𝑡𝑎𝑏𝑙𝑒 𝑜𝑛 𝑡ℎ𝑒 𝑙𝑒𝑓𝑡
2. Concept 𝐴𝑠𝑠𝑖𝑔𝑛 𝑡𝑜 𝑒𝑣𝑒𝑟𝑦 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑎 𝑚𝑢𝑙𝑡𝑖𝑝𝑙𝑒 𝑣𝑎𝑙𝑢𝑒 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑙𝑢𝑚𝑛. A 12
B 13
C 14
D 21
E 23
F 24
G 26
H 28
I 31
J K 32 34
L 36
M 39
N 41
O 42
P 43
Q 46
R 48
S 62
T 63
U 64
V 68
W 69
X Y 82 84
Z 86
′ 93
_ 96
X
Y 123 243 363
𝐴 = 12 = 421 = 631 = 841 ∴ 𝐴 𝑠 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒 𝑖𝑠 𝑎𝑛𝑦 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑣𝑎𝑙𝑢𝑒𝑠 {421, 631, 841}, 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 12 𝑎𝑝𝑝𝑒𝑎𝑟𝑒𝑑 3 𝑡𝑖𝑚𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑙𝑢𝑚𝑛. 𝑤ℎ𝑖𝑙𝑒 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑖𝑛𝑔 𝑡ℎ𝑒 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝐴 𝑦𝑜𝑢 𝑎𝑟𝑒 𝑓𝑟𝑒𝑒 𝑡𝑜 𝑐ℎ𝑜𝑜𝑠𝑒 𝑎𝑛𝑦 𝑜𝑛𝑒 𝑜𝑓 𝑡ℎ𝑜𝑠𝑒 3 𝑣𝑎𝑙𝑢𝑒𝑠. ′
𝑇ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑡𝑎𝑏𝑙𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑠 𝑒𝑣𝑒𝑟𝑦 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑤𝑖𝑡ℎ 𝑖𝑡 ′ 𝑠 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 A 421 631 841
B 621 931
C 821
D 241 361 481
E 641 961
F 211 632 842
G 311 932
H
I
J
411
261 391
461 691
K 861
L 212 422 843
M 312 622
N 281
O 121 362 482
P
Q
681
321 962
R 213 423 633
S 131 392
T 122 242 483
U 231 692
V
W
431
322 642
141
𝐋𝐞𝐭′𝐬 𝐞𝐧𝐜𝐫𝐲𝐩𝐭 𝐭𝐡𝐞 𝐬𝐞𝐧𝐭𝐞𝐧𝐜𝐞 "𝐈′𝐦_𝐅𝐢𝐧𝐞" 𝑇ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑡𝑎𝑏𝑙𝑒 𝑡𝑒𝑙𝑙 𝑢𝑠 ℎ𝑜𝑤 𝑚𝑎𝑛𝑦 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟 𝑡ℎ𝑎𝑡 𝑤𝑒 𝑎𝑟𝑒 𝑓𝑟𝑒𝑒 𝑡𝑜 𝑐ℎ𝑜𝑜𝑠𝑒 𝑎𝑛𝑦 𝑜𝑛𝑒 𝑜𝑓 𝐼 ′ 𝑀 2 2 2
_ 2
𝐹 3
𝐼 𝑁 2 1
𝐸 2
∴ 𝑇ℎ𝑒 𝑠𝑒𝑛𝑡𝑒𝑛𝑐𝑒 "I'm_Fine" 𝑐𝑎𝑛 𝑏𝑒 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑒𝑑 𝑖𝑛 192 (2 ∗ 2 ∗ 2 ∗ 2 ∗ 3 ∗ 2 ∗ 1 ∗ 2) 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡 𝑤𝑎𝑦𝑠. 𝑇ℎ𝑒 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑡𝑎𝑏𝑙𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑡ℎ𝑎𝑡 𝑖 𝑐ℎ𝑜𝑠𝑒 𝑟𝑎𝑛𝑑𝑜𝑚𝑙𝑦 𝑡𝑜 𝑒𝑛𝑐𝑟𝑦𝑝𝑡 𝑡ℎ𝑒 𝑠𝑒𝑛𝑡𝑒𝑛𝑐𝑒 "I'm_Fine". 𝐼 ′ 𝑀 _ 𝐹 𝐼 𝑁 𝐸 261 132 622 462 842 391 281 961 ∴ (I'm_Fine = 261132622462842391281961) 𝑖𝑛 𝑐𝑖𝑝ℎ𝑒𝑟. 𝑜𝑟 𝑤𝑒 𝑐𝑎𝑛 𝑐ℎ𝑜𝑜𝑠𝑒 𝑠𝑜𝑚𝑒 𝑜𝑡ℎ𝑒𝑟 𝑣𝑎𝑙𝑢𝑒𝑠, 𝑙𝑖𝑘𝑒 𝐼 ′ 𝑀 _ 𝐹 𝐼 𝑁 𝐸 391 132 312 462 842 261 281 641 ∴ (I'm_Fine = 𝟑𝟗𝟏𝟏𝟑𝟐𝟑𝟏𝟐𝟒𝟔𝟐𝟖𝟒𝟐𝟐𝟔𝟏𝟐𝟖𝟏𝟔𝟒𝟏) 𝒊𝒏 𝒄𝒊𝒑𝒉𝒆𝒓.
Z
′
_
341
132 262
232 462
𝐋𝐞𝐭′𝐬 𝐝𝐞𝐜𝐫𝐲𝐩𝐭 𝐭𝐡𝐞 𝐜𝐢𝐩𝐡𝐞𝐫 𝟖𝟒𝟏𝟐𝟒𝟐𝟒𝟖𝟑𝟔𝟑𝟏𝟖𝟐𝟏𝟖𝟔𝟏𝟐𝟑𝟐𝟒𝟐𝟏𝟐𝟏𝟏𝟏𝟐𝟐𝟗𝟔𝟏𝟒𝟐𝟑𝟐𝟖𝟏𝟑𝟔𝟐𝟒𝟖𝟐𝟐𝟖𝟏 𝑭𝒊𝒓𝒔𝒕, 𝑆𝑒𝑝𝑎𝑟𝑎𝑡𝑒 𝑡ℎ𝑒𝑚 841 242 483 631 821 861 232 421 211 122 961 423 281 362 482 281 𝑺𝒆𝒄𝒐𝒏𝒅, 𝑇𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚 𝑡ℎ𝑒𝑚 𝑡𝑜 𝑡ℎ𝑒 𝑚𝑎𝑡ℎ𝑒𝑚𝑎𝑡𝑖𝑐𝑎𝑙 𝑜𝑝𝑒𝑟𝑎𝑡𝑜𝑟 𝑓𝑜𝑟𝑚 𝑡ℎ𝑎𝑡 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑠 𝑡ℎ𝑒 𝐸𝑆𝑀 𝑠𝑒𝑡 𝑎𝑛𝑑 𝑡ℎ𝑒 𝑖𝑛𝑑𝑒𝑥 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑖𝑛 𝑡ℎ𝑎𝑡 𝑠𝑒𝑡 841 242 483 631 821 861 232 421 211 122 961 423 281 362 482 281 𝑻𝒉𝒊𝒓𝒅, 𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑜𝑠𝑒 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠, 𝑡𝑜 𝑏𝑒 12
63 63 12 14 34 96
12 24 63 23 48
41 42 42 41
𝑭𝒐𝒓𝒕𝒉, 𝐸𝑥𝑐ℎ𝑎𝑛𝑔𝑒 𝑒𝑣𝑒𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 𝑤𝑖𝑡ℎ 𝑖𝑡𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟. 𝐴
𝑇
𝑇
𝐴
𝐶
𝐾
_ 𝐴
𝐹
𝑇
𝐸
𝑅
𝑁
𝑂
𝑂
𝑁
𝐴𝑡𝑡𝑎𝑐𝑘_𝐴𝑓𝑡𝑒𝑟𝑛𝑜𝑜𝑛 ∴ 𝟖𝟒𝟏 𝟐𝟒𝟐 𝟒𝟖𝟑 𝟔𝟑𝟏 𝟖𝟐𝟏 𝟖𝟔𝟏 𝟐𝟑𝟐 𝟒𝟐𝟏 𝟐𝟏𝟏 𝟏𝟐𝟐 𝟗𝟔𝟏 𝟒𝟐𝟑 𝟐𝟖𝟏 𝟑𝟔𝟐 𝟒𝟖𝟐 𝟐𝟖𝟏 = Attack_Afternoon
3. Encryption in Real Life 𝐼𝑛 𝑟𝑒𝑎𝑙 𝑙𝑖𝑓𝑒 𝑐𝑖𝑝ℎ𝑒𝑟𝑠, 𝐼 𝑒𝑛𝑐𝑜𝑢𝑟𝑎𝑔𝑒 𝑢𝑠𝑖𝑛𝑔 𝑡ℎ𝑒 𝐸𝑆𝑀 𝑜𝑓 𝑡ℎ𝑒 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑛 = {1000 ≤ 𝑛 ≤ 9999 | 𝑛 ∈ 𝑁}, 𝑜𝑟 𝑛 = {100000 ≤ 𝑛 ≤ 999999 | 𝑛 ∈ 𝑁}, 𝑜𝑟 𝑏𝑖𝑔𝑔𝑒𝑟, 𝑤ℎ𝑖𝑐ℎ 𝑖𝑠 𝑚𝑜𝑟𝑒 𝑠𝑎𝑓𝑒𝑟. 𝑊𝑖𝑡ℎ 𝑡ℎ𝑖𝑠 ℎ𝑢𝑔𝑒 𝑎𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑤𝑒 𝑤𝑜𝑛′ 𝑡 𝑎𝑠𝑠𝑖𝑔𝑛 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑡𝑜 𝑒𝑣𝑒𝑟𝑦 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟, 𝑏𝑢𝑡 𝑎 𝑤𝑖𝑑𝑒 𝑟𝑎𝑛𝑔𝑒 𝑜𝑓 𝑛𝑢𝑚𝑏𝑒𝑟𝑠, 𝑤ℎ𝑒𝑟𝑒 𝑒𝑣𝑒𝑟𝑦 𝑛𝑢𝑚𝑏𝑒𝑟 ℎ𝑎𝑠 𝑎 𝑤𝑖𝑑𝑒 𝑟𝑎𝑛𝑔𝑒 𝑜𝑓 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠, 𝑒𝑛𝑑𝑖𝑛𝑔 𝑖𝑛 ℎ𝑎𝑣𝑖𝑛𝑔 𝑎 𝑤𝑖𝑑𝑒𝑟 𝑟𝑎𝑛𝑔𝑒 𝑜𝑓 𝑒𝑛𝑐𝑟𝑦𝑝𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒𝑠 𝑓𝑜𝑟 𝑒𝑣𝑒𝑟𝑦 𝑐ℎ𝑎𝑟𝑎𝑐𝑡𝑒𝑟.
4. References -
𝑇ℎ𝑒 𝑆𝑒𝑡 𝑜𝑓 𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑆𝑤𝑎𝑝𝑝𝑒𝑑 𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠 𝑆𝑜𝑢𝑟𝑐𝑒: ℎ𝑡𝑡𝑝𝑠://𝑤𝑤𝑤. 𝑎𝑐𝑎𝑑𝑒𝑚𝑖𝑎. 𝑒𝑑𝑢/16131410 /𝑇ℎ𝑒_𝑆𝑒𝑡_𝑜𝑓_𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡_𝑆𝑤𝑎𝑝𝑝𝑒𝑑_𝑀𝑢𝑙𝑡𝑖𝑝𝑙𝑒𝑠_𝐸𝑆𝑀_
