1 Montgomery Modular Multiplication in Hard- ware

FEI KEMT
Algorithm 4 – 2 Modular addition
Require: Two integers x, y < 2n
Ensure: Sum z = x + y mod 2n
1: z ⇐ x + y
2: T ⇐ z − 2n
3: if T ≥ 0 then
4: z ⇐ T
5: end if
6: return z
Algorithm 4 – 3 Modular subtraction
Require: Two integers x, y < 2n
Ensure: Difference z = x − y mod 2n
1: T = z ⇐ x − y
2: if z < 0 then
3: z ⇐ T + 2n
4: end if
5: return z
4.2.4 Parallelization of the Algorithm
ECM can be perfectly parallelized by using different curves in parallel since the
computations of each unit are completely independent. For the control of more
than one ECM unit, it is essential to know that both phases, phase 1 and phase 2,
are controlled completely identically, independent of the composite to be factored.
Solely the curve parameter and possibly the modulus of the units and, hence, the
coordinates of the initial point differ. Thus, all units have to be initialized differently
which is done by simply writing the values into the corresponding memory locations
sequentially.
During the execution of both phases, exactly the same commands can be sent to
all units in parallel. Since the runtime of multiplication/squaring is constant (does
not rely on input values) and for addition/subtraction differs at most in 2(e + 1)
clock cycles, all units can execute the same command in approximately the same
time.
After phase 2, the results are read from the units one after another. The required
time for this data I/O is negligible for one ECM unit since the computation time of
both phases dominates. For several units in parallel, the computation time does not
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