A New Cryptosystem and Algebraic Constructions for its Key Space K. T. Arasu, Riverside Research Beavercreek, Ohio karasu@RiversideResearch.org
Snake-and-Ladder Blocks We introduce a new symmetric cryptosystem based on snake-and- ladder blocks. We call it the “snakes -and- ladders cryptography system” (SALCS). Our cryptographic system relies on the fact that subsequent encryptions depend on the previous encryptions. • Ladder Block: This is a block formed by a number of message units which is equal to the magnitude of the corresponding sub-key. The ladder block is formed by using the message units starting from the current position in the forward direction or simply saying the encryption takes place in forward direction. • Snake Block: This is a block formed by a number of message units which is equal to the magnitude of the corresponding sub-key. The snake block is formed by using the message units starting from the current position in the reverse direction or simply saying the encryption takes place in backward/reverse direction. 2
We assume that the message to be transmitted is a plaintext with 𝑜 values, which is represented by a vector of length 𝑜 . If the message has more than 𝑜 values than the procedure, it is repeated as many times as needed. If it has less than 𝑜 letters, then we pad the plaintext with the letter “space” a sufficient number of times. Encryption of each block (Message Unit) is in itself independent of the previous blocks (Message Units), but the input to the individual block (i.e. the block itself) is dependent on the previous blocks and key. The decryption follows the same algorithm as the encryption but in reverse order. This feature of the key makes the whole system incredibly secure while using a very moderate sized keyspace. 3
Figure 1: The snake-and-ladder blocks work iteratively together to encrypt the plaintext message 4
General Encryption/Decryption Algorithm • Let us write the message to be encrypted as 𝒏 = 𝑛 1 𝑛 2 … … … … . . 𝑛 𝑜 . Here each 𝑛 𝑗 is a message unit, which is a 32-bit integer. The private encryption key is a vector 𝐿 , of length 𝑚 , 𝐿 = Є 1 𝐿 1 , Є 2 𝐿 2 , … , Є 𝑚 𝐿 𝑚 where Є 𝑗 ∈ {−1, +1} , and 𝐿 𝑗 is any positive integer that represents the number of message units to be encrypted at the 𝑢ℎ encryption step. 𝑗 5
General Encryption/Decryption Algorithm • Define the two sets 𝑄 and 𝑂 as follows: 𝑄 = {𝑗|Є 𝑗 = 1}; 𝑂 = {𝑗|Є 𝑗 = −1} . As will become clear once the entire encryption process is described, we have the arithmetic constraint 𝑜 = [𝐿 𝑗 − |𝑄| + 1] . We also need for each 𝑗𝜗𝑄 𝑗 ∈ 𝑂, 𝐿 𝑗 ≤ 𝐿 𝑘 = 𝑇 𝑗 ; where this summation runs over all 𝑘 satisfying 𝑘 ∈ 𝑄, 𝑘 < 𝑗 . 6
Explanation of the Encryption/Decryption Process • For 𝑗 = 1 , the first block (𝑛 1 , 𝑛 2 , … , 𝑛 𝐿1 ) is encrypted using the key 𝐿 1 . • For 𝑗 > 1 , suppose that the first (𝑗 − 1) blocks have been encrypted as 𝑌 1 , 𝑌 2 , … … … … , 𝑌 . 𝑇 𝑗 • If Є 𝑗 = 1 , the i th ladder block to be encrypted is 𝑌 𝑇 𝑗 𝑛 𝑇 𝑗 +1 𝑛 𝑇 𝑗 +2 … 𝑛 𝑇 𝑗 +𝑙 𝑗 −1 using the key 𝐿 𝑗 . • If Є 𝑗 = −1 , the i th snake block to be encrypted is 𝑌 𝑇 𝑗 −(𝐿 𝑗 −1) … 𝑌 𝑇 𝑗 −1 𝑌 using the key 𝐿 𝑗 . 𝑇 𝑗 7
Explanation of the Encryption/Decryption Process • Note that the ladder (resp. snake) portion of our scheme can be compared to Shannon’s concept of confusion (resp. diffusion). • The decryption of the cipher text received can be carried out by using the key in the same manner as in encryption, but in reversed order, i.e. decryption with the last key is done first, followed by the second last and so on up to the first key. • Upon receiving the cipher text 𝑫 = 𝐷 _1 𝐷 _2… 𝐷 _ 𝑜 , the decrypter applies the same encryption algorithm with the securely received private key ( 𝐿 _1, 𝐿 _2,…, 𝐿 _ 𝑚 ), performing at the ith iteration level (while decrypting), the same operation as the encryption during the ( 𝑚 − 𝑗 +1)^ 𝑡𝑢 iteration level. 8
Encryption and Decryption via Difference Set Pairs Let 𝐻 be an additively written abelian group of order 𝑤 . Let 𝐵 and 𝐶 be subsets of 𝐻 of size 𝑙 and 𝑙′ respectively. Let 𝐵 ∩ 𝐶 = 𝑓 . If the list/multiset of differences (𝑦 − 𝑧: 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶, 𝑦 ≠ 𝑧) contains each nonzero element exactly 𝜇 times, then we call (𝐵, 𝐶) a difference set pair (DSP) in 𝐻 with parameters (𝑤, 𝑙, 𝑙′, 𝑓, 𝜇) . 9
The proposed encryption scheme proceeds as follows: • The message m to be sent is converted into a series of integers by replacing every 32 bits in a bit-stream with their respective integer value. • The sets 𝐵 and 𝐶 of a (𝑤, 𝑙, 𝑙 ′ , 𝑓, 𝜇) DSP can be represented as a 𝑤 × 𝑤 matrix using the group action 𝐻 where the first rows are determined by the sets as 𝐵 (respectively 𝐶 ) by the indicator function: −1 𝑗𝑔 ∈ 𝐵 +1 𝑝𝑢ℎ𝑓𝑠𝑥𝑗𝑡𝑓 10
• The (𝑤 + 1) × (𝑤 + 1) encryption matrix 𝐹 is obtained from the 𝑤 × 𝑤 matrix 𝐵 by adding an extra row and column full of 1’s (we denote 1 by + below). i.e. 𝐹 = + + 𝑤+1 × 𝑤+1 + 𝐵 • Similarly, we define the decryption matrix as 𝐸 = + + 𝑤+1 × 𝑤+1 + 𝐶 • In the block matrices 𝐹 and 𝐸 above, we misuse the letters 𝐵 and 𝐶 for the 𝑤 × 𝑤 matrices defined via the sets 𝐵 and 𝐶 as explained above. 11
Case I : r = -1 We only discuss the case when 𝒔 = −𝟐 (the case 𝑠 = 0 would need slight modifications). 𝑈 , Now, we calculate 𝐹𝐸 𝑤 − 2𝑙 ′ + 1 𝑤 − 2𝑙 ′ + 1 𝑤 + 1 … 𝑤 − 2𝑙 + 1 4 𝑓 − 𝜇 + 𝑠 + 1 𝑠 + 1 𝐹𝐸 𝑈 = ⋮ 𝑠 + 1 ⋱ 𝑠 + 1 𝑤 − 2𝑙 + 1 𝑠 + 1 4 𝑓 − 𝜇 + 𝑠 + 1 where we recall that, 𝑠 = 𝑤 − 2 𝑙 + 𝑙′ + 1 + 4 𝜇 , is called the correlation constant. We apply this scheme for the case 𝑠 = −1 and 𝑙 = (𝑤 + 1)/2 (the so- called “balanced” case). Thus, 𝑤 − 2𝑙 ′ + 1 𝑤 − 2𝑙 ′ + 1 𝑤 + 1 … 0 4 𝑓 − 𝜇 0 𝐹𝐸 𝑈 = . ⋮ 0 ⋱ 0 0 0 4 𝑓 − 𝜇 12
Encryption Algorithm Let 𝒏 = (𝑛 1 , 𝑛 2 , … , 𝑛 𝑤+1 ) be the plaintext. The ciphertext 𝑫 is then given by 𝑫 = 𝒏 ∙ 𝐹 = 𝒏 ∙ 𝑫 1 𝐹 , 𝒏 ∙ 𝑫 2 𝐹 , … , 𝒏 ∙ 𝑫 𝑤+1 𝐹 Thus, the 𝑗 𝑢ℎ component of 𝑫 is the dot product of 𝒋 𝐹 is the 𝑗 𝑢ℎ the vectors 𝒏 and 𝑫 𝒋 𝐹 where 𝑫 column vector of the encryption matrix 𝐹 . 13
Decryption Algorithm We use the corresponding decryption matrix 𝐸 to decrypt the ciphertext 𝑫 : We first compute the vector 𝒆 as follows: 𝒆 = 𝑫 ∙ 𝑫 1 𝐸 𝑈 , 𝑫 ∙ 𝑫 2 𝐸 𝑈 , … , 𝑫 ∙ 𝑫 𝑤+1 𝐸 𝑈 . Thus, the 𝑗 𝑢ℎ component of 𝒆 is the dot product of 𝑫 and 𝑫 𝒋 𝐸 𝑈 , where 𝑫 𝒋 𝐸 𝑈 is the 𝑗 𝑢ℎ column vector of the matrix 𝐸 𝑈 . 14
• The original plaintext 𝒏 is recovered as follows: 𝒏 1 = 𝒏 1 = 𝒆 1 𝑤 + 1 1 𝒏 𝑗 = 𝒏 𝑗 = 4(e− λ) [𝒆 𝑗 − 𝒏 1 (𝑤 − 2𝑙 ′ + 1) for i > 1 • This 𝒏 is the final decrypted message which is same as the original message. 15
• Case II: r = 0 • We proceed as in case I, but now we use v x v matrices A and B without bordering using an extra row and column of plus ones. Similar calculations work in this case. 16
• It is important to note that this proposed system adheres to Kerckhoff’s principle. That is, our cryptosystem is demonstrably secure even though the entirety of the algorithm may be disclosed publicly. Our strength and security lies solely in the vast complexity of the keyspace. • Table 1: Key Space Sizes (Here l is length of the key vector K) 17
Table 1: Key Space Sizes (Here l is length of the key vector K) 𝑚 100^ 𝑚 250^ 𝑚 300^ 𝑚 400^ 𝑚 20 10 40 10 47 10 50 10 52 10 60 10 71 10 74 10 78 30 10 80 10 95 10 99 10 104 40 10 100 10 119 10 124 10 130 50 100 10 200 10 238 10 248 10 260 18
Algebraic Approach for Key Generation • Since our cryptosystem of Section 3 hinges on the keyspace, we develop an algebraic mechanism to construct the keyspace. Toward this, we now introduce the ideas needed to formulate the theory of binary array pairs. We study them via their combinatorial counterparts: difference set pairs in abelian groups. • Let 𝐻 be an additively written abelian group of order 𝑤 . Let 𝐵 and 𝐶 be subsets of 𝐻 of size 𝑙 and 𝑙′ respectively. Let 𝐵 ∩ 𝐶 = 𝑓 . If the list/multiset of differences (𝑦 − 𝑧: 𝑦 ∈ 𝐵, 𝑧 ∈ 𝐶, 𝑦 ≠ 𝑧) contains each nonzero element exactly 𝜇 times, then we call (𝐵, 𝐶) a difference set pair (DSP) in 𝐻 with parameters (𝑤, 𝑙, 𝑙′, 𝑓, 𝜇) . 19
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