Arjen Markus 19 november 2002. In reaction to a discussion in the Tclers' chat room:

A function that is used quite regularly in cryptographic applications is:

P(n,r,m) = n^r mod m

(n^r denotes n to the power r, all three parameters are positive integers)

This function does not exist as such within the expr command, but it is simple to create a procedure that evaluates this function efficiently.

The parameters for this function are usually large enough to make a direct calculation impossible. But the trick is to recognise that n^r can be written as n^(r/2)*n^(r/2) when r is even. If r is odd, then write it as:

n^r = n * (n^(r-1))

and then we have the first case back.

Furthermore, the modulo function is very well behaved:

(h * g) mod m = ( (h mod m) * (g mod m) ) mod m

Given these observations, here is a recursive proc that evaluates the above function in at most 2*log2(r) steps. This can probably be improved, but I am a bit lazy.

AM Repaired a stupid mistake at the bottom. rmax provided some useful comments - so changed the body into a if-elseif-chain

**WARNING** The code as shown *only* works correctly on 64-bits machines or when all intermediate results fit in a 32-bits integer. This requires some more work right now ...

AMG: Is this warning still applicable in recent versions of Tcl?

AM: No, from 8.5 onwards, Tcl has arbitrary-precision integers, so that even 52131^53124 can be computed (even though it is a ridiculously large number). The code and comments should be updated.

proc powm {n r m} { # # Take care of the trivial cases first (they also stop the recursion) # if { $r == 0 } { set result 1 } elseif { $r == 1 } { set result [expr {$n % $m}] } elseif { $r%2 == 0 } { set nn [powm $n [expr {$r/2}] $m] set result [expr { ($nn*$nn) % $m} ] } else { set nn [powm $n [expr {$r-1}] $m] set result [expr { ($n*$nn) % $m} ] } return $result } # # Testing the function # # The following cases are easy to verify manually: # powm(1,10,2) = 1 # powm(2,10,3) = 1024 mod 3 = 1 # powm(3,4,5) = 81 mod 5 = 1 puts "powm(1,10,2) = [powm 1 10 2 ]" puts "powm(2,10,3) = [powm 2 10 3 ]" puts "powm(3, 4,5) = [powm 3 4 5 ]" # # Larger numbers (results respectively, 1, 71, 381, 12685) # puts "powm(31,24,15) = [powm 31 24 15 ]" puts "powm(131,124,115) = [powm 131 124 115 ]" puts "powm(2131,3124,4115) = [powm 2131 3124 4115 ]" puts "powm(52131,53124,54115) = [powm 52131 53124 54115 ]"

Note that in the last test cases the intermediate numbers would be much too big to deal with directly:

52131^53124 = Order(10^246334)

or, to put it in another way, a figure with a quarter million digits

*rmax* -- If you want to use this function in real-world applications you might be interested in the following optimized version which runs 20..50% faster, depending on the Tcl version. Interpretation is left as an exercise to the reader.

proc powm {n r m} { expr {$r & 1 ? $n * [powm $n [expr {$r-1}] $m] % $m : $r ? [set nn [powm $n [expr {$r >> 1}] $m]] * $nn % $m : 1 } }

When you have Tcl 8.4 you can use the wide() function to promote n to a wide integer which raises the 32bit barrier noted above to 64bit:

proc powm_wide {n r m} { powm [expr {wide($n)}] $r $m }