Finite Field Arithmetic

This section describes the Intel IPP primitives that implement arithmetic operations with elements of finite fields [ANT]. Arithmetic of the following finite fields is implemented:

G(p)	A finite field of p elements represented by integers modulo p, where p is an odd prime number. This field is also known as a prime finite field.
GF(p^d)	A finite field of pd elements represented by equivalence classes modulo g(x) of polynomials whose coefficients belong to GF(p), where g(x) is an irreducible polynomial of degree d. Coefficients of g(x) are elements of the GF(p) field. This field is also known as an extension field of degree d of GF(p) or the Galois field.
GF(p^d^2)	A quadratic extension of GF(p^d).

Table “Intel IPP Finite Field Arithmetic Functions” lists all the finite field arithmetic functions.

Each element E of GF(p) is represented by an unsigned big number, which, in turn, is represented by a data array Ipp32u pe[length], so that

Each element E of GF(p^d) is represented by a polynomial of degree less than d, which, in turn, is represented by an array of coefficients pe[d] that belong to GF(p).

Each element E of GF(p^d^2) is represented by a polynomial of degree less than 2, which, in turn, is represented by an array of coefficients pe[2] that belong to GF(p^d).

For polynomials that represent elements of both fields, a coefficient of a lower degree is stored in an element of the respective array with a smaller index.

The Intel IPP finite field arithmetic functions use context structures of the following types to carry data of the field and field element:

GF(p)	IppsGFPState and IppsGFPElement, respectively.
GF(p^d)	IppsGFPXState and IppsGFPXElement, respectively.
GF(p^d^2)	IppsGFPXQState and IppsGFPXQElement, respectively.

Comparison functions GFPCmpElement, GFPXCmpElement, and GFPXQCmpElement return the result of comparison:

typedef enum {
    IppsElementEQ = 0,	// elements are equal
    IppsElementNE = 1,	// elements are not equal
    IppsElementGT = 2,	// the first element is greater than the second one
    IppsElementLT = 3,	// the first element is less than the second one
    IppsElementNA = 4	// elements are not comparable
} IppsElementCmpResult; 
 

• Functions for the GF(p) Field
  
• Functions for the GF(p^d) Field
  
• Functions for the GF(p^d^2) Field
  

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