This file offers some guidance for developers who need to create assembly 
language Macros for an unsupported processor to support the Comba or KCM 
methods for modular multiplication. Note that the "standard" C build of MIRACL 
may be fast enough, or else the provided C macros may be sufficient. 

An .mcs file contains C or assembly language macros to implement some simple
operations on big number digits. On most modern Load/Store RISC processors
the optimal assembly language required is fortunately quite generic - and  
that's why this approach is possible. The Instruction set need only support 
unsigned multiply, add and subtract. It should also support standard indexed 
addressing modes, where the effective memory address is calculated by adding 
an offset to a register. The compiler should ideally support in-line assembly.

Correct memory address offsets and label names are automatically inserted by 
the mex utility. It uses the standard C function "fprintf" to expand the
macros, and hence inserts appropriate numbers where-ever %d appears in the 
macro. This implies that if the symbol % should be part of assembly language 
syntax, it must be replaced by %%.

These macros when extracted by the mex.c program from the specified .mcs 
file are used to implement some fast algorithms.

MULTIPLY

                a2    a1    a0
                b2    b1    b0
                --------------
             a2.b0 a1.b0 a0.b0
       a2.b1 a1.b1 a0.b1
 a2.b2 a1.b2 a0.b2
 ------------------------------
    c4    c3    c2    c1   c0


Here (a1.b0) is a partial product. As we add up each column, the result is
accumulated in a triple-precision register x|y|z. When the column is totalled
z is written to memory and 0|x|y becomes the "carry" to the next column. Note
that x will always be "small", less than the number of partial products in
the longest column.

The algorithm for fast multi-precision multiplication multiplies a*b=c and
requires the following macros.

        MUL_START

        Push any registers if necessary, Initialise pointers to a, b and c, 
        and zero the triple-precision register required to accumulate the 
        partial products.

        STEP

        Calculate a partial product and add it to the triple-register.
        The array offsets from the pointers to a and b are inserted 
        automatically by the mex utility.

        MFIN

        Store total for this column z, and calculate the carry for the next.

        MUL_END

        Store the left over carry in the left-most column, and pop registers
        if necessary.


MULTUP

Here we just need the first (lower) half of c=a*b; So we use the same 
MUL_START, STEP, and MFIN macros as above. However no carry is needed from 
the last column, so a simpler LAST macro is employed here.

SQUARE

Multiprecision squaring can be done nearly twice as fast, as partial products
appear twice in most columns, but only need to be calculated once. Observe
that if a=b in the above example then a2.b0 = a0.b2 in the third column.

This algorithm finds c=a*a

        SQR_START

        Push registers, form pointers to a and c, zeroise triple register 
        
        DSTEP

        Calculate a partial product and add it twice to column total in
        the triple register.

        SELF

        Calculate a "diagonal element", e.g. (a1.a1) and add to total.

        SFIN

        Store total for this column z, and calculate the carry for the next 

        SQR_END

        Store the left over carry in the left-most column, and pop registers
        if necessary.

REDC

Montgomery's modular reduction algorithm calculates a%=b; It also accesses
the pre-computed variable "ndash".

        REDC_START

        Push registers, get pointers to a and b, and put ndash in a register.
        Initialise the triple register x|y|z to 0|0|a[0]

        RFINU

        Multiply ndash*z and store lower half of result in a[i] for i-th
        column. Multiply this number now by b[0]. Add the result to the 
        triple register x|y|z. Set the triple register to reflect the 
        "carry" to the next column 0|x|y. Add a[i+1] to the triple register.

        RFIND

        Store z into a[i]. Set triple register to reflect the "carry"
        to the next column. Add a[i+1] to the triple register.

        REDC_END

        Store the left-over carry in the left-most two columns.

This algorithm also uses the STEP macro as described above.

ADDITION

This algorithm does a simple element-by-element adition c=a+b, 
propagating the carries.

        ADD_START

        Gets pointers to a, b and c. Adds the first two elements 
        c[0]=a[0]+b[0]

        ADD

        Adds-with-carry c[i]=a[i]+b[i]

        ADD_END

        If the carry flag is still set, set the variable "carry"=1, 
        otherwise =0

INCREMENT

Very similar to the above, but this time calculates a+=b

        INC_START

        Gets pointers to a and b. Adds the first elements a[0]+=b[0]

        INC

        Adds-with-carry a[i]+=b[i]

        INC_END

        If the carry flag is still set, set the variable "carry=1", 
        otherwise 0

SUBTRACTION

Find c=a-b, propagating borrows

        SUB_START

        Gets pointers to a, b and c. Subtracts the first two elements 
        c[0]=a[0]-b[0]

        SUB

        Subtracts-with-borrow c[i]=a[i]-b[i]

        SUB_END

        If there is a "borrow" outstanding, set the variable "carry"=1, 
        otherwise =0. **NOTE** This MAY be indicated by carry flag=1 OR
        by carry flag=0 - it depends on architecture, so be careful.

DECREMENT

Find a-=b, propagating borrows

        DEC_START

        Gets pointers to a and b. Subtracts the first two elements 
        a[0]-=b[0]

        DEC

        Subtracts-with-borrow a[i]-=b[i]

        DEC_END

        If there is a "borrow" outstanding, set the variable "carry"=1, 
        otherwise =0. **NOTE** This MAY be indicated by carry flag=1 OR
        by carry flag=0 - it depends on architecture, so be careful.

SUMMATION

Adds c=a+b in a "for" loop. Each time around the loop a fixed block of
digits are added. A total of n such blocks are to be added.

        KADD_START

        Initialise pointers to a, b and c. Move n into a register. Set
        the carry flag to zero. Provide a label for looping back to. Note
        that label numbers are automatically inserted by the mex utility.

        KASL

        Decrement the n register. If its zero jump to label at the end of 
        this macro. If its not, advance the pointer registers to point to
        the next block, and branch back to label in KADD_START. 
        **NOTE** It is vitally important that this macro does NOT affect 
        the carry flag

        KADD_END

        If the carry flag is still set, set the variable "carry=1", 
        otherwise 0

This algorithm also uses the ADD macro. See above


INCREMENTATION

Adds a+=b in a "for" loop. Each time around the loop a fixed block of
digits are added. A total of n such blocks are to be added.

        KINC_START

        Initialise pointers to a and b. Move n into a register. Set
        the carry flag to zero. Provide a label for looping back to. Note
        that label numbers are automatically inserted by the mex utility.

        KIDL

        Decrement the n register. If its zero jump to label at the end of 
        this macro. If its not, advance the pointer registers to point to
        the next block, and branch back to label in KINC_START. 
        **NOTE** It is vitally important that this macro does NOT affect 
        the carry flag

        KINC_END

        If the carry flag is still set, set the variable "carry=1", 
        otherwise 0

This algorithm also uses the INC macro. See above


DECREMENTATION

Subtracts a-=b in a "for" loop. Each time around the loop a fixed block of
digits are subtracted. A total of n such blocks are to be subtracted.

        KDEC_START

        Initialise pointers to a and b. Move n into a register. Set
        the carry flag to that state which indicates no borrow. Provide a 
        label for looping back to. Note that label numbers are automatically 
        inserted by the mex utility.

        KDEC_END

        Set the variable "carry"  to 1 if the carry flag is in that state
        which reflects an outstanding "borrow", otherwise 0

This algorithm also uses the KIDL and DEC macros. See above.


NEW! - April 2002

Interleaved steps can be used in multiplication to allow for improved 
instruction scheduling. This could be a lot faster if the multiply unit takes 
more than 1 clock cycle. The idea is to expose more ILP (Instruction Level 
Parallelism) for a modern pipelined (and possibly super-scaler) load-store 
processor to chew on.

In the calculation of the sum of partial products in a column, replace

 STEPM - Multiply
 STEPA - Add
 STEPM - Multiply
 STEPA - Add
 STEPM - Multiply
 STEPA - Add
 STEPM - Multiply
 STEPA - Add

with the interleaved

 STEP1M - Multiply 1

 STEP2M - Multiply 2
 STEP1A - Add 1
 STEP1M - Multiply 1
 STEP2A - Add 2
 STEP2M - Multiply 2
 STEP1A - Add 1

 STEP2A - Add 2

The same applies to DSTEP for squaring.

In this way the multiply instruction gets more time to complete to its 
destination registers.

If the MEX program sees that STEP1M macro is present as well as STEP, 
it will permit scheduling with the -s flag as above. Of course this requires 
that there to be enough registers - STEP1x and STEP2x should ideally use 
different registers. 

If this is going to help it is particularly important that the destination 
registers of the multiply steps STEP1M and STEP2M be distinct.

Many processors use hardware dynamic scheduling (like the Pentium II). 
For such a processor scheduling the code like this will have little 
effect. The ARM assembler re-orders the code automatically for 
optimum scheduling, so again scheduling the code like this will have little 
impact.

Some example *.mcs files in this format are included - see c.mcs and
arm.mcs for example.

This idea could be extended in a fairly obvious way, for example

 STEP1M - Multiply 1
 STEP2M - Multiply 2
 STEP3M - Multiply 3
 
 STEP1A - Add 1
 STEP1M - Multiply 1
 STEP2A - Add 2
 STEP2M - Multiply 2
 STEP3A - Add 3
 STEP3M - Multiply 3
 STEP1A - Add 1
 STEP1M - Multiply 1

 STEP2A - Add 2
 STEP3A - Add 3
 STEP1A - Add 1

This is currently NOT supported, it would require a relatively simple 
modification to mex.c

This might be justified in the case of a very deeply pipelined multiplier.
The idea would be that Multiply 1, 2, and 3 might all be active 
simultaneously.

