Tracing the First Full Round of IDEA Algorithm

IDEA (International Data Encryption Algorithm) is a symmetric block cipher that operates on 64-bit plaintext blocks using a 128-bit key, performing 8 identical rounds with operations: multiplication modulo $2^{16}+1$, addition modulo $2^{16}$, and XOR.

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Given Data

• Key subkeys (4-bit each): $K_1 = 1100$, $K_2 = 1010$, $K_3 = 0000$, $K_4 = 1111$, $K_5 = 0101$, $K_6 = 1001$
• Input text blocks (4-bit each): $X_1 = 1011$, $X_2 = 1110$, $X_3 = 1011$, $X_4 = 1000$

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IDEA First Round Structure

In each round, 6 subkeys are used. The operations in one full round are:

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Step-by-Step Trace

Step A: Initial Four Operations

• i) $X_1 \odot K_1$ (Multiply mod $2^4 + 1 = 17$)

$$X_1 \odot K_1 = 1011 \times 1100 \mod 17 = 11 \times 12 \mod 17 = 132 \mod 17 = 132 - 7(17) = 132 - 119 = 13 = 1101$$

• ii) $X_2 \boxplus K_2$ (Add mod $2^4 = 16$)

$$X_2 \boxplus K_2 = 1110 + 1010 \mod 16 = 14 + 10 \mod 16 = 24 \mod 16 = 8 = 1000$$

• iii) $X_3 \boxplus K_3$ (Add mod $2^4 = 16$)

$$X_3 \boxplus K_3 = 1011 + 0000 \mod 16 = 11 + 0 \mod 16 = 11 = 1011$$

• iv) $X_4 \odot K_4$ (Multiply mod $2^4 + 1 = 17$)

$$X_4 \odot K_4 = 1000 \times 1111 \mod 17 = 8 \times 15 \mod 17 = 120 \mod 17 = 120 - 7(17) = 120 - 119 = 1 = 0001$$

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Step B: XOR Operations

• v) Step(i) $\oplus$ Step(iii):

$$1101 \oplus 1011 = 0110$$

• vi) Step(ii) $\oplus$ Step(iv):

$$1000 \oplus 0001 = 1001$$

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Step C: MA (Multiplication-Addition) Structure

• vii) Step(v) $\odot K_5$ (Multiply mod 17):

$$0110 \times 0101 \mod 17 = 6 \times 5 \mod 17 = 30 \mod 17 = 13 = 1101$$

• viii) Step(vi) $\boxplus$ Step(vii) (Add mod 16):

$$1001 + 1101 \mod 16 = 9 + 13 \mod 16 = 22 \mod 16 = 6 = 0110$$

• ix) Step(viii) $\odot K_6$ (Multiply mod 17):

$$0110 \times 1001 \mod 17 = 6 \times 9 \mod 17 = 54 \mod 17 = 54 - 3(17) = 54 - 51 = 3 = 0011$$

• x) Step(vii) $\boxplus$ Step(ix) (Add mod 16):

$$1101 + 0011 \mod 16 = 13 + 3 \mod 16 = 16 \mod 16 = 0 = 0000$$

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Step D: Final XOR Operations

• xi) Step(i) $\oplus$ Step(ix): $1101 \oplus 0011 = 1110$
• xii) Step(iii) $\oplus$ Step(ix): $1011 \oplus 0011 = 1000$
• xiii) Step(ii) $\oplus$ Step(x): $1000 \oplus 0000 = 1000$
• xiv) Step(iv) $\oplus$ Step(x): $0001 \oplus 0000 = 0001$

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Output of First Round

After swapping the inner two blocks (Step xii and Step xiii):

$$\text{Output} = 1110 \quad | \quad 1000 \quad | \quad 1000 \quad | \quad 0001$$

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Conclusion: The first full round of IDEA uses all 6 subkeys and combines three algebraically incompatible operations (XOR, addition mod $2^n$, multiplication mod $2^n+1$) to achieve strong diffusion and confusion in a single round itself.