Author

T h a m i r A l s h a m m a r i
tha2341[at]rit[.]edu

What is BSCA?

Binary Sequence Cumulative Algorithm (BSCA) is a mathematical process that draws a unique relation between binary strings and integers. BSCA introduces a new decimal representation of binary other than the power of 2. The power of a BSCA process is based on the addition mathematical operation. The BSCA function accepts a binary string as input and generates two unique relatively prime integers as output, called the Binary Pair (A, B).

BSCA Function

Let A represents the zeros and B represents the ones. Given an arbitrary binary string S of length N ≥ 1, there exist two unique relatively prime integers A,B Є ℤ if we initiate A₀, B₀ with 1 and cumulatively exchange the sum of A_i , B_i based on their round positions in S_i :

BSCA Properties

• a_n ≠ b_n
• a_n , b_n > 0
• GCD⁡(a_n , b_n) = 1
• There exists multiplicative inverse d_n such that d_nb_n ≡ 1 mod a_n
• Min⁡ value of a_n , b_n = 1
• Max⁡ value of a_n , b_n is computed by Binet's Formula:

BSCA Test

Choose Random Bits Length (n)

8 (2^3)16 (2^4)32 (2^5)64 (2^6)128 (2^7)256 (2^8)512 (2^9)1024 (2^10)2048 (2^11)4096 (2^12)8192 (2^13)16384 (2^14)32768 (2^15)65536 (2^16)

 

• 8 (2^3)
• 16 (2^4)
• 32 (2^5)
• 64 (2^6)
• 128 (2^7)
• 256 (2^8)
• 512 (2^9)
• 1024 (2^10)
• 2048 (2^11)
• 4096 (2^12)
• 8192 (2^13)
• 16384 (2^14)
• 32768 (2^15)
• 65536 (2^16)

Loading

Run

Run

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Outputs

Runtime(s_n) =

Runtime(a_n) =
Length in Bits = 1
Length in Digits = 1

1

Runtime(b_n) =
Length in Bits = 1
Length in Digits = 1

1

Runtime(d_n) =
Length in Bits = 0
Length in Digits = 1

0

Runtime GCD(a_n , b_n) =

0