Bit level equivalence of unsigned and two-s complement multiplication

• Let $\vec{x}$ and $\vec{y}$ be bit vectors of length $w$
• Define $x$ and $y$ as the values represented by these bits in two’s complement form
  • $x=B2T_w(\vec{x})$ and $B2T_w(\vec{y})$
• Let $x^{\prime }$ and $y^{\prime }$ as the values represented by the bits $\vec{x}$ and $\vec{y}$ in unsigned form
  • $x^{\prime }=B2U_w(\vec{x})$ and $y^{\prime }=B2U(\vec{y})$

$$T2B_w(x{\ast }_w^{t}y)=U2B(x^{\prime }{\ast }_w^u{y}^{\prime })$$

Demonstration

Sources

• Computer Systems - A Programmer’s Perspective

Related Notes

• Multiplying signed integers
• Multiplying unsigned integers