Carry-less Multiplication (CLMUL) is an extension to the x86 instruction set used by microprocessors from Intel and AMD which was proposed by Intel in March 2008^[1] and made available in the Intel Westmere processors announced in early 2010. Mathematically, the instruction implements multiplication of polynomials over the finite field GF(2) where the bitstring ${\displaystyle a_{0}a_{1}\ldots a_{63}}$ represents the polynomial ${\displaystyle a_{0}+a_{1}X+a_{2}X^{2}+\cdots +a_{63}X^{63}}$. The CLMUL instruction also allows a more efficient implementation of the closely related multiplication of larger finite fields GF(2^k) than the traditional instruction set.^[2]

One use of these instructions is to improve the speed of applications doing block cipher encryption in Galois/Counter Mode, which depends on finite field GF(2^k) multiplication. Another application is the fast calculation of CRC values,^[3] including those used to implement the LZ77 sliding window DEFLATE algorithm in zlib and pngcrush.^[4]

ARMv8 also has a version of CLMUL. SPARC calls their version XMULX, for "XOR multiplication".

The instruction computes the 128-bit carry-less product of two 64-bit values. The destination is a 128-bit XMM register. The source may be another XMM register or memory. An immediate operand specifies which halves of the 128-bit operands are multiplied. Mnemonics specifying specific values of the immediate operand are also defined:

A EVEX vectorized version (VPCLMULQDQ) is seen in AVX-512.

• Intel
  • Westmere processor (March 2010).
  • Sandy Bridge processor
  • Ivy Bridge processor
  • Haswell processor
  • Broadwell processor (with increased throughput and lower latency^[5])
  • Skylake (and later) processor
  • Goldmont processor
• AMD:
  • Jaguar-based processors and newer ^[6]
  • Puma-based processors and newer
  • "Heavy Equipment" processors
    • Bulldozer-based processors ^[7]
    • Piledriver-based processors
    • Steamroller-based processors
    • Excavator-based processors and newer
  • Zen processors
  • Zen+ processors
  • Zen2 (and later) processors

The presence of the CLMUL instruction set can be checked by testing one of the CPU feature bits.

• Finite field arithmetic
• AES instruction set
• FMA3 instruction set
• FMA4 instruction set
• AVX instruction set