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Multiplication algorithms are systematic methods used to multiply two numbers, especially in digital electronics and computer arithmetic.
Ensure efficiency when dealing with large datasets (e.g., cryptography, simulations).
Implemented as circuit designs in CPUs, GPUs, and DSPs to make multiplication fast and power-efficient.
🔍 Different algorithms are optimized for different use cases, balancing speed, power consumption, and hardware complexity.
Binary multiplication follows the same principles as decimal multiplication, except it only uses 0 and 1.
Each bit of the multiplier is ANDed with the multiplicand.
Each new partial product is shifted left according to its bit position.
Add up all the shifted partial products.
📌 Example: Multiply 1011 (11 in decimal) × 1101 (13 in decimal):
This is the foundation of all advanced algorithms.
Designed to handle signed binary multiplication more efficiently than naive methods.
Instead of multiplying bit-by-bit, it looks at two adjacent bits of the multiplier to decide whether to add, subtract, or skip the multiplicand.
Store multiplicand (M), multiplier (Q), accumulator (A = 0).
Look at two bits: (current bit of Q and the bit to its right).
Right-shift the register pair (A, Q).
Repeat for all bits.
Reduces the number of addition/subtraction operations.
Multiplying signed integers in CPUs.
Improves Booth's by looking at three bits at a time (instead of two).
3-bit Groups
Recoding
Multiply
Group the multiplier into overlapping 3-bit groups.
Translate these into values like -2, -1, 0, +1, +2.
Multiply and accumulate partial products based on these recoded digits.
Reduces the number of steps by halving the number of partial products.
High-speed processors, floating-point units.
A combinational circuit that directly implements binary multiplication using AND gates and adders.
AND Gates
Shifting
Adders
Using AND gates.
Align them by shifting (based on bit position).
Add them using a grid of adders.
📌 Example: For 4-bit numbers, it creates a 4×4 AND gate array, with adders handling partial sums.
Simple design, parallel generation of partial products.
Slower for large numbers, consumes more hardware resources.
Small multipliers in microcontrollers.
An optimized parallel approach to speed up the summation of partial products.
Partial Products
Tree Reduction
Final Addition
Generate all partial products.
Use compressor circuits (half and full adders) in a tree-like structure to reduce them to just two rows.
Perform a final addition using a fast adder (like a carry-lookahead adder).
Much faster than an Array Multiplier because it reduces addition levels (logarithmic depth).
More complex circuit design.
High-speed CPUs, GPUs, and DSP chips.
Each algorithm has its own hardware design style:
Decides when to add, subtract, or skip.
Arithmetic circuits.
For right-shift operations.
Generate partial products.
Sum them in parallel.
(3:2, 4:2 adders) → Reduce partial products quickly.
Produces the final product.
RSA/ECC use very large multiplications (FFT-based or Booth variants).
Filters, FFT, convolution → Wallace tree & Booth multipliers.
Floating-point multipliers (Modified Booth).
Simple array multipliers.
💡 The choice of multiplication algorithm depends on the specific requirements of the application, balancing speed, power consumption, and hardware complexity.
🚀 Each algorithm has its strengths and is chosen based on the specific requirements of speed, power, and hardware complexity in different computing applications.