4.4 Coding-Decoding, Cipher Patterns & Symbol Operations

Coding-Decoding, Cipher Patterns & Symbol Operations

Coding-decoding and symbol manipulation items on the ACET evaluate symbolic processing agility, rule application, and working memory. These questions present encrypted words, artificial language translations, or novel mathematical operators. Rather than relying on trial-and-error, you must approach cipher breakdown systematically using position mapping, morphological alignment, and operator evaluation formulas.

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1. Classical Cipher Classification & Decryption Rules

In cryptography items, plaintext letters are transformed into ciphertext letters or numbers via deterministic encoding algorithms.

1. Caesar Shift & Variable-Shift Ciphers

Every letter is shifted forward or backward in the alphabet by a specified value $k$:

$C_i = (P_i + k - 1 \pmod{26}) + 1$

• Constant Shift: Every letter moves by $+k$ (e.g., $+3 \implies A \rightarrow D, B \rightarrow E$).
• Progressive Shift: The shift increases arithmetically per letter position: $+1, +2, +3, +4, \dots$
  • Plaintext HELL $\rightarrow H(+1)E(+2)L(+3)L(+4) \implies I, G, O, P$.
• Alternating Shift: Alternates between $+k_1$ and $-k_2$ (e.g., $+2, -1, +2, -1$).

2. Atbash Complementary Cipher (Reverse Alphabetic Mapping)

Each letter is mapped to its complementary opposite position in the 26-letter alphabet:

$\text{Position}(C_i) + \text{Position}(P_i) = 27$

• Mapping pairs: $A \leftrightarrow Z, B \leftrightarrow Y, C \leftrightarrow X, D \leftrightarrow W, \dots, M \leftrightarrow N$.
• Plaintext CAT $\implies C(3) \rightarrow X(24), A(1) \rightarrow Z(26), T(20) \rightarrow G(7) \implies XZG`.

3. Block Reversal & Structural Transposition

Transposition ciphers rearrange letter order without altering letter identities:

• Word Halving: Split word into two equal halves and reverse each half.
  • MONKEY $\rightarrow MON \mid KEY \rightarrow NOM \mid YEK \implies NOMYEK`.
• Odd-Even Index Swap: Swap adjacent letter pairs: $P_1 \leftrightarrow P_2, P_3 \leftrightarrow P_4$.
  • MATH $\rightarrow AMHT`.

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2. Letter-to-Number Encoding Systems

Number coding maps letters to numerical values based on explicit algebraic formulas.

                 LETTER-TO-NUMBER CODING SCHEMES
                                |
        +-----------------------+-----------------------+
        |                                               |
        v                                               v
  Direct Index Sum                               Weighted / Scaled Code
(Sum of A=1...Z=26)                           (Vowels ×2, Consonants +1)
  Example: CAT                                  Example: CAT
  C=3, A=1, T=20                                C(4) + A(2) + T(21)
  Sum = 3+1+20 = 24                             Sum = 4+2+21 = 27

Common ACET Numeric Encoding Formulas

1. Direct Positional Sum: $\text{Code} = \sum_{i=1}^n \text{Pos}(L_i)$.
2. Opposite Positional Sum: $\text{Code} = \sum_{i=1}^n (27 - \text{Pos}(L_i))$.
3. Length Multiplied Index: $\text{Code} = \left(\sum \text{Pos}(L_i)\right) \times \text{Word Length}$.

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3. Decoding Artificial Languages (Synthetic Syntax)

Artificial language questions present translated word pairs and require decoding unknown phrases. Use the Morphological Alignment Algorithm to isolate vocabulary:

Step-by-Step Translation Algorithm

Given the synthetic phrases:

1. "pina loma" = "blue house"

2. "pina kuta" = "blue car"

3. "tivo kuta" = "fast car"

4. Isolate Overlapping Words:

   • Compare Phrase 1 & Phrase 2: The word "pina" appears in both synthetic strings, and "blue" appears in both English meanings. $\implies$ "pina" = "blue".
   • Compare Phrase 2 & Phrase 3: The word "kuta" appears in both synthetic strings, and "car" appears in both English meanings. $\implies$ "kuta" = "car".

5. Deduce Remaining Vocabulary:

   • From Phrase 1: Since "pina" = "blue", then "loma" = "house".
   • From Phrase 3: Since "kuta" = "car", then "tivo" = "fast".

6. Determine Structural Syntax (Modifier Order):

   • In "pina loma" ("blue house"), "pina" (modifier) comes first, followed by "loma" (noun). Syntax order matches English (Modifier + Noun).

7. Synthesize Target Phrase:

   • To translate "fast house": Modifier ("fast" = "tivo") + Noun ("house" = "loma") $\implies$ "tivo loma".

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4. Custom Symbol Operations & Non-Standard Algebra

ACET symbol operation items invent custom arithmetic operators defined by algebraic expressions. Treat these as function evaluations.

Evaluation Rules

• Definition: $a \otimes b = a^2 - 3b + 1$
• Non-Commutative Property: $a \otimes b \neq b \otimes a$. Always preserve left/right variable placement.
• Order of Precedence: Evaluate expressions inside parentheses first.

Nested Symbol Walkthrough

Evaluate $(4 \otimes 2) \otimes 3$ using $a \otimes b = a^2 - 3b + 1$:

1. Step 1: Evaluate Inner Bracket $(4 \otimes 2)$:

   • Let $a = 4$ and $b = 2$.
   • $4 \otimes 2 = (4)^2 - 3(2) + 1 = 16 - 6 + 1 = 11$.

2. Step 2: Evaluate Outer Expression $(11 \otimes 3)$:

   • Let $a = 11$ and $b = 3$.
   • $11 \otimes 3 = (11)^2 - 3(3) + 1 = 121 - 9 + 1 = 113$.

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5. Rapid Cryptanalytic Inspection Summary