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In mathematics and computer science, Horner's method (or Horner's scheme) is an algorithm for polynomial evaluation.Although named after William George Horner, this method is much older, as it has been attributed to Joseph-Louis Lagrange by Horner himself, and can be traced back many hundreds of years to Chinese and Persian mathematicians. [1]
The definition of matrix multiplication is that if C = AB for an n × m matrix A and an m × p matrix B, then C is an n × p matrix with entries = =. From this, a simple algorithm can be constructed which loops over the indices i from 1 through n and j from 1 through p, computing the above using a nested loop:
Here we consider operations over polynomials and n denotes their degree; for the coefficients we use a unit-cost model, ignoring the number of bits in a number. In practice this means that we assume them to be machine integers.
A multiplication uses one carryless multiply to produce a product (up to 2n − 1 bits), another carryless multiply of a pre-computed inverse of the field polynomial to produce a quotient = ⌊product / (field polynomial)⌋, a multiply of the quotient by the field polynomial, then an xor: result = product ⊕ ((field polynomial) ⌊product ...
The optimal number of field operations needed to multiply two square n × n matrices up to constant factors is still unknown. This is a major open question in theoretical computer science. As of January 2024, the best bound on the asymptotic complexity of a matrix multiplication algorithm is O(n 2.371552).
Polynomial factoring algorithms use basic polynomial operations such as products, divisions, gcd, powers of one polynomial modulo another, etc. A multiplication of two polynomials of degree at most n can be done in O(n 2) operations in F q using "classical" arithmetic, or in O(nlog(n) log(log(n)) ) operations in F q using "fast" arithmetic.
The elements of GF(2 n), i.e. a finite field whose order is a power of two, are usually represented as polynomials in GF(2)[X]. Multiplication of two such field elements consists of multiplication of the corresponding polynomials, followed by a reduction with respect to some irreducible polynomial which is taken from the construction of the field.
Perform a carry-less multiplication of two 64-bit polynomials over the finite field GF(2)[X]. PCLMULLQLQDQ xmmreg,xmmrm [rm: 66 0f 3a 44 /r 00] Multiply the low halves of the two registers. PCLMULHQLQDQ xmmreg,xmmrm [rm: 66 0f 3a 44 /r 01] Multiply the high half of the destination register by the low half of the source register.