43.3.4.9 Full Multiply

The purpose of this service is to multiply two large numbers, X and Y, and optionally accumulate/subtract from a third large number, Z, returning the result, R.

The available options are as follows:

• Work in the GF(2ⁿ) field or in the standard arithmetic field
• Add of a supplemental CarryOperand
• Overlap of the operands is possible, taking into account some constraints
• Modular Reduction of the computation result (see Modular Reduction)

This service provides the following (if not computing a modular reduction of the result):

R = [Z] ± (X × Y + CarryOperand)

Or (if computing a modular reduction of the result):

R = ([Z] ± (X × Y + CarryOperand))mod N

The service name for this operation is Fmult.

In these computations, the following data has to be provided:

• R the result (pointed by {nu1RBase,u2Xlength +u2YLength})
• X one input number or GF(2ⁿ) polynomial (pointed by{nu1XBase,u2XLength})
• Y one input number or GF(2ⁿ) polynomial (pointed by{nu1YBase,u2YLength})
• Z one optional input number or GF(2n) polynomial (pointed by {nu1ZBase,u2Xlength +u2YLength})
• CarryOperand (provided through the Carry Options and Carry values)

If using the big modular reduction option, the Multiply operation is followed by a reduction (see Modular Reduction). In this case, the length of Cns is 64 bytes.

If using the modular reduction option, the Multiply operation is followed by a reduction (see Modular Reduction). In this case the following parameters must be additionally provided:

• N—the modulus (pointed by {nu1ModBase,u2Modlength +4})
• Cns—the reduction constant
  • In case of Big reduction, Cns is pointed by {nu1CnsBase,64bytes}.
  • In case of Fast or Normalized reduction, Cns is pointed by (pointed by {nu1CnsBase,u2ModLength+ 8})

For example, if u2ModLength, u2XLength and u2YLength are 0x80 bytes, the length of the R space is 2*(u2ModLength + 4) = 0x108 bytes because of the constraints of modular reduction.

In case of Fast or Normalized Reduction, the length of the result is u2ModLength + 4 so 0x84 bytes. Thus, the zone X has a length of 0x84 bytes (at least). The multiplication of X by Y provides a result of length 0x100 bytes in the zone R so the 8 MSB bytes must be previously padded with zero bytes (in offsets 0x100 to 0x107).

The options are set by the u2Options input parameter, which is composed of:

• the mandatory Full Multiplication operation option described in Table 43-26
• the mandatory CarryOperand option described in Table 43-27 and Table 43-28
• the facultative Modular Reduction option (see Modular Reduction). If the Modular Reduction is not requested, this option is absent.

The u2Options number is calculated by an Inclusive OR of the options.

Some Examples in C language are:

• Operation: Full Multiply only without carry and without Modular Reduction

  PUKCL(u2Options) = SET_MULTIPLIEROPTION(PUKCL_FMULT_ONLY) | SET_CARRYOPTION(CARRY_NONE);

• Operation: Full Multiply with addition with Specific/CarryIn addition and with Fast Modular Reduction

  PUKCL(u2Options) = SET_MULTIPLIEROPTION(PUKCL_FMULT_ADD) |

  SET_CARRYOPTION(ADD_CARRY) |

  PUKCL_REDMOD_REDUCTION |

  PUKCL_REDMOD_USING_FASTRED;

The following table shows all of the necessary parameters for the Full Multiply option. When the Addition or Subtraction option is not chosen, it is not necessary to fill in the nu1ZBase parameter.

PUKCL_PARAM PUKCLParam;
PPUKCL_PARAM pvPUKCLParam = &PUKCLParam;


// Gf2n and CarryIn shall be beforehand filled (with zero or one) 
PUKCL(Specific).Gf2n = ...;
PUKCL(Specific).CarryIn = ...;

PUKCL(u2Option) =...;
// Depending on the option specified, not all fields should be filled 
PUKCL_Fmult(nu1XBase) = <Base of the ram location of X>; 
PUKCL_Fmult(u2XLength) = <Length of X>;
PUKCL_Fmult(nu1YBase) = <Base of the ram location of Y>; 
PUKCL_Fmult(u2YLength) = <Length of Y>; 
PUKCL_Fmult(nu1ZBase) = <Base of the ram location of Z>; 
PUKCL_Fmult(nu1RBase) = <Base of the ram location of R>;

// vPUKCL_Process() is a macro command, which populates the service name
// and then calls the library... 
vPUKCL_Process(Fmult,pvPUKCLParam); 
if (PUKCL(u2Status) == PUKCL_OK)
            {
            // The Full multiply has been executed correctly
            ...
            }
else // Manage the error

The following conditions must be avoided to ensure that the service works correctly:

• nu1XBase, nu1YBase, nu1RBase or nu1ZBase are not aligned on 32-bit boundaries
• {nu1XBase, u2XLength}, {nu1YLength, u2YLength}, {nu1ZBase, u2XLength+u2YLength} or{nu1RBase, u2XLength+u2YLength} are not in Crypto RAM
• u2XLength, u2YLength is either: < 4, > 0xffc or not a 32-bit length
• {nu1RBase, u2XLength+u2YLength} overlaps {nu1YBase, u2YLength} or{nu1RBase, u2XLength+u2YLength} overlaps {nu1XBase, u2XLength}
• {nu1RBase, u2XLength+u2YLength} overlaps {nu1ZBase, u2XLength+u2YLength} and nu1RBase> nu1ZBase

If a modular reduction is specified, the relevant parameters must be defined according to the chosen reduction and follow the description in Modular Reduction. Additional constraints to be respected and error codes are described in this section and in Table 43-49.