Modular Arithmetic

Mon, May 11, 2020 02:00 CEST

Authors: Aurélien Ooms
Tags: Number Theory, Arithmetic, Algebra

Positional notation.

An arbitrary precision integer on a computer is represented as a list of digits (limbs). They are just bigger. Today’s computers can hold 64 bits per limb.

Modular arithmetic consists in applying addition and multiplication to integers modulo a certain number \(n\).

Given an implementation of addition, multiplication, and division over the integers, we can emulate addition or multiplication modulo \(n\) by performing the corresponding operation on integers then taking the result modulo \(n\).

However, taking the modulo is usually more expensive than multiplication: the schoolbook algorithm for division takes quadratic time and the divide and conquer solution [1] solves the problem with up to \(\log n\) input size multiplications: one with the schoolbook method, two with Karatsuba multiplication, roughly 2.6 with Toom-Cook \(k=3\), \(\log n / 2\) with \(n \log n\)-time multiplication, and \(\log n\) with linear-time multiplication (no algorithm is known to achieve this).

To be fair, for a certain range of input sizes, this divide and conquer approach to division is faster than the methods we are presenting here, in particular for small input sizes that should be handled with the schoolbook multiplication method or Karatsuba’s algorithm.

Note that depending on the base being used to represent our numbers, some division and modulo operations can be done amazingly fast: when representing numbers in base 10 for instance, that number modulo 10 is simply its last digit.

We are interested in a technique called Montgomery’s trick. This technique shows that, as long as \(gcd(R,N) = 1\) with \(R>N\) it is possible to trade operations modulo \(N\) for operations modulo \(R\). Hence, if modulo \(R\) can be done efficiently, so can modulo \(N\).

The Trick

If \(gcd(R,N)=1\) then there exists integers R' and N' such that \(RR' + NN' = 1\) (Bezout’s identity). Then \(NN' = 1 - RR'\) which means \(N'\) is the inverse of \(N\) modulo \(R\).

Given an input \(T \in [0, RN-1]\), informally, we can compute the pseudo-quotient \(q = T/N \mod R\) by \(q = TN' \mod R\). Then the pseudo-rest of this pseudo-division by \(N \mod R\) is \(r = T - qN = T - TNN' = 0 \mod R\).

\[ \begin{aligned} q &= ((T \mod R) N') \mod R \\ r &= TR' \mod N \\ &= TR' - (qR')N \\ &= (T - qN) R' = (T - qN) / R. \end{aligned} \]

where the last line works since \(r = 0 \mod R\).

Hence multiplication by \(R' \mod N\) can be achieved by keeping the high digit of \(T - qN\) in base \(R\):

  • \(q = (T \mod R) N' \mod R\),
  • \(r = (T - qN) / R\).

where \(\mod R\) corresponds to taking the lower digits and \(/R\) corresponds to taking the higher digits. This involves only two multiplications:

  • \((T \mod R) N'\), and
  • \(qN\).

Note that \(q \le R-1\), hence \(T - qN \in [-(R-1)N, RN-1]\), hence \(r\) is in the range \([-(R-1)N/R, N-1/R]\) and since \(r\) is an integer this range collapses to \([1-N, N-1]\). Hence, \(r\) can be brought down to the representation range \([0,N-1]\) with one addition.

Time Complexity

The trick uses two integer multiplications: one to compute \((T \mod R) N'\), and one to compute \(qN\). Note that all operands are smaller than \(R\). If \(R\) is made out of \(k\) digits, then these multiplications are all \(k \times k\) multiplications.

Note that in the first multiplication we only care about the lower order bits of the result, and in the second multiplication we only care about the higher order bits of the result. This can be exploited to further reduce the overhead of this method for relatively small inputs.

Addition and Multiplication modulo \(N\)

To implement addition and multiplication easily we convert numbers \(A\) and \(B\) to \(AR \mod N\) and \(BR \mod N\) using the standard division algorithm. This incurs an initial conversion cost that is negligible when lots of arithmetic operations are performed (for instance, in modular exponentiation).

The addition \(A+B\) corresponds to \((A+B)R \mod N = AR \mod N + BR \mod N\) where the last \(\mod N\) operations can be done with a single subtraction.

The multiplication \(A*B\) corresponds to \((A*B)R \mod N = (AR \mod N)(BR \mod N) R' \mod N\) where the multiplication by \(R' \mod N\) is achieved through Montgomery’s trick.