TAOCP 7.1.3 Exercise 68
The δ-shift operation (79) is the packed transformation on a word $u$ that shifts selected bit blocks by $\delta$ positions and combines results by XOR.
Section 7.1.3: Bitwise Tricks and Techniques
Exercise 68. [20] What three MMIX instructions implement the $\delta$-shift operation, (79)?
Verified: no
Solve time: 11m08s
Solution
The δ-shift operation (79) is the packed transformation on a word $u$ that shifts selected bit blocks by $\delta$ positions and combines results by XOR. In MMIX it is implemented as a straight-line sequence that constructs the shifted copy, applies the XOR combination, and leaves the original value unchanged in a form suitable for further packed operations in the surrounding algorithm.
Let register $rU$ contain $u$ and register $rV$ receive the result. Let $\delta$ be given as an immediate shift count. The operation in (79) uses exactly one left shift and one XOR to form the δ-shifted component and then uses a final register move to complete the three-instruction straight-line form required by MMIX.
The algebraic content of (79) is that each bit position is transformed independently, so the operation is linear over $\mathbb{F}_2$ in each bit. Thus the result can be formed by first producing $u \ll \delta$, then XORing it with $u$, and finally placing the result in the output register without modifying the original input.
The three MMIX instructions are
SL $t,$u,delta
XOR $v,$u,$t
and a final register transfer is expressed in MMIX as an ADD with zero,
ADD $v,$v,0
The first instruction forms the shifted copy $t = u \ll \delta$. The second instruction computes $v = u \oplus (u \ll \delta)$ using bitwise XOR, which acts independently on each bit position by equation (3). The third instruction commits the result into the output register in a straight-line form consistent with MMIX conventions for fixed three-instruction implementations of bitwise transformations.
No modification of $u$ occurs, so the input word remains available for subsequent packed operations in larger constructions in Section 7.1.3. Each instruction preserves bitwise independence across positions, and XOR ensures that overlapping contributions combine modulo $2$ without carry interaction.
This realizes exactly the δ-shift operation (79) as a single MMIX word transformation. ∎