Exponentiation by squaring

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In mathematics and computer programming, exponentiating by squaring is a general method for fast computation of large positive integer powers of a number, or more generally of an element of a semigroup, like a polynomial or a square matrix. Some variants are commonly referred to as square-and-multiply algorithms or binary exponentiation. These can be of quite general use, for example in modular arithmetic or powering of matrices. For semigroups for which additive notation is commonly used, like elliptic curves used in cryptography, this method is also referred to as double-and-add.

Basic method[edit]

The method is based on the observation that, for a positive integer n, we have

This method uses the bits of the exponent to determine which powers are computed.

This example shows how to compute using this method. The exponent, 13, is 1101 in binary. The bits are used in left to right order. The exponent has 4 bits, so there are 4 iterations.

First, initialize the result to 1: .

Step 1) ; bit 1 = 1, so compute ;
Step 2) ; bit 2 = 1, so compute ;
Step 3) ; bit 3 = 0, so we are done with this step (equivalently, this corresponds to );
Step 4) ; bit 4 = 1, so compute .

If we write in binary as , then this is equivalent to defining a sequence by letting and then defining for , where will equal .

This may be implemented as the following recursive algorithm:

  Function exp_by_squaring(x, n)
    if n < 0  then return exp_by_squaring(1 / x, -n);
    else if n = 0  then return  1;
    else if n = 1  then return  x ;
    else if n is even  then return exp_by_squaring(x * x,  n / 2);
    else if n is odd  then return x * exp_by_squaring(x * x, (n - 1) / 2);

Although not tail-recursive, this algorithm may be rewritten into a tail recursive algorithm by introducing an auxiliary function:

  Function exp_by_squaring(x, n)
    return exp_by_squaring2(1, x, n)
  Function exp_by_squaring2(y, x, n)
    if n < 0  then return exp_by_squaring2(y, 1 / x, - n);
    else if n = 0  then return  y;
    else if n = 1  then return  x * y;
    else if n is even  then return exp_by_squaring2(y, x * x,  n / 2);
    else if n is odd  then return exp_by_squaring2(x * y, x * x, (n - 1) / 2).

A tail-recursive variant may also be constructed using a pair of accumulators instead of an auxiliary function as seen in the F# example below. The accumulators a1 and a2 can be thought of as storing the values and where i and j are initialized to 1 and 0 respectively. In the even case i is doubled, and in the odd case j is increased by i. The final result is where .

let exp_by_squaring x n =
    let rec _exp x n' a1 a2 =
        if   n' = 0   then 1
        elif n' = 1   then a1*a2
        elif n'%2 = 0 then _exp x (n'/2) (a1*a1) a2
        else               _exp x (n'-1) a1 (a1*a2)
    _exp x n x 1

The iterative version of the algorithm also uses a bounded auxiliary space, and is given by

  Function exp_by_squaring_iterative(x, n)
    if n < 0 then
      x := 1 / x;
      n := -n;
    if n = 0 then return 1
    y := 1;
    while n > 1 do
      if n is even then 
        x := x * x;
        n := n / 2;
      else
        y := x * y;
        x := x * x;
        n := (n  1) / 2;
    return x * y

Computational complexity[edit]

A brief analysis shows that such an algorithm uses squarings and at most multiplications, where denotes the floor function. More precisely, the number of multiplications is one less than the number of ones present in the binary expansion of n. For n greater than about 4 this is computationally more efficient than naively multiplying the base with itself repeatedly.

Each squaring results in approximately double the number of digits of the previous, and so, if multiplication of two d-digit numbers is implemented in O(dk) operations for some fixed k, then the complexity of computing xn is given by

2k-ary method[edit]

This algorithm calculates the value of xn after expanding the exponent in base 2k. It was first proposed by Brauer in 1939. In the algorithm below we make use of the following function f(0) = (k, 0) and f(m) = (s, u), where m = u·2s with u odd.

Algorithm:

Input
An element x of G, a parameter k > 0, a non-negative integer n = (nl−1, nl−2, ..., n0)2k and the precomputed values .
Output
The element xn in G
y := 1; i := l - 1
while i ≥ 0 do
    (s, u) := f(ni)
    for j := 1 to k - s do
        y := y2 
    y := y * xu
    for j := 1 to s do
        y := y2
    i := i - 1
return y

For optimal efficiency, k should be the smallest integer satisfying[1][clarification needed]

Sliding-window method[edit]

This method is an efficient variant of the 2k-ary method. For example, to calculate the exponent 398, which has binary expansion (110 001 110)2, we take a window of length 3 using the 2k-ary method algorithm and calculate 1, x3, x6, x12, x24, x48, x49, x98, x99, x198, x199, x398. But, we can also compute 1, x3, x6, x12, x24, x48, x96, x192, x199, x398, which saves one multiplication and amounts to evaluating (110 001 110)2

Here is the general algorithm:

Algorithm:

Input
An element x of G, a non negative integer n=(nl−1, nl−2, ..., n0)2, a parameter k > 0 and the pre-computed values .
Output
The element xnG.

Algorithm:

y := 1; i := l - 1
while i > -1 do
    if ni = 0 then
        y := y2' i := i - 1
    else
        s := max{i - k + 1, 0}
        while ns = 0 do
            s := s + 1[notes 1]
        for h := 1 to i - s + 1 do
            y := y2
        u := (ni, ni-1, ..., ns)2
        y := y * xu
        i := s - 1
return y

Montgomery's ladder technique[edit]

Many algorithms for exponentiation do not provide defence against side-channel attacks. Namely, an attacker observing the sequence of squarings and multiplications can (partially) recover the exponent involved in the computation. This is a problem if the exponent should remain secret, as with many public-key cryptosystems. A technique called "Montgomery's ladder"[2] addresses this concern.

Given the binary expansion of a positive, non-zero integer n = (nk−1...n0)2 with nk−1 = 1, we can compute xn as follows:

x1 = x; x2 = x2
for i = k - 2 to 0 do
    If ni = 0 then
        x2 = x1 * x2; x1 = x12
    else
        x1 = x1 * x2; x2 = x22
return x1

The algorithm performs a fixed sequence of operations (up to log n): a multiplication and squaring takes place for each bit in the exponent, regardless of the bit's specific value. A similar algorithm for multiplication by doubling exists.

This specific implementation of Montgomery's ladder is not yet protected against cache timing attacks: memory access latencies might still be observable to an attacker, as different variables are accessed depending on the value of bits of the secret exponent. Modern cryptographic implementations use a "scatter" technique to make sure the processor always misses the faster cache.[3]

Fixed-base exponent[edit]

There are several methods which can be employed to calculate xn when the base is fixed and the exponent varies. As one can see, precomputations play a key role in these algorithms.

Yao's method[edit]

Yao's method is orthogonal to the 2k-ary method where the exponent is expanded in radix b = 2k and the computation is as performed in the algorithm above. Let n, ni, b, and bi be integers.

Let the exponent n be written as

where for all .

Let xi = xbi.

Then the algorithm uses the equality

Given the element x of G, and the exponent n written in the above form, along with the precomputed values xb0...xbw−1, the element xn is calculated using the algorithm below:

y = 1, u = 1, j = h - 1
while j > 0 do
    for i = 0 to w - 1 do
        if ni = j then
            u = u × xbi
    y = y × u
    j = j - 1
return y

If we set h = 2k and bi = hi, then the ni values are simply the digits of n in base h. Yao's method collects in u first those xi that appear to the highest power ; in the next round those with power are collected in u as well etc. The variable y is multiplied times with the initial u, times with the next highest powers, and so on. The algorithm uses multiplications, and elements must be stored to compute xn.[1]

Euclidean method[edit]

The Euclidean method was first introduced in Efficient exponentiation using precomputation and vector addition chains by P.D Rooij.

This method for computing in group G, where n is a natural integer, whose algorithm is given below, is using the following equality recursively:

where . In other words, a Euclidean division of the exponent n1 by n0 is used to return a quotient q and a rest n1 mod n0.

Given the base element x in group G, and the exponent written as in Yao's method, the element is calculated using precomputed values and then the algorithm below.

Begin loop   
    Find , such that .
    Find , such that .
    Break loop if .
    Let , and then let .
    Compute recursively , and then let .
End loop;
Return .

The algorithm first finds the largest value among the ni and then the supremum within the set of { ni \ iM }. Then it raises xM to the power q, multiplies this value with xN, and then assigns xN the result of this computation and nM the value nM modulo nN.

Further applications[edit]

The same idea allows fast computation of large exponents modulo a number. Especially in cryptography, it is useful to compute powers in a ring of integers modulo q. It can also be used to compute integer powers in a group, using the rule

Power(x, −n) = (Power(x, n))−1.

The method works in every semigroup and is often used to compute powers of matrices.

For example, the evaluation of

13789722341 (mod 2345) = 2029

would take a very long time and much storage space if the naïve method were used: compute 13789722341, then take the remainder when divided by 2345. Even using a more effective method will take a long time: square 13789, take the remainder when divided by 2345, multiply the result by 13789, and so on. This will take less than modular multiplications.

Applying above exp-by-squaring algorithm, with "*" interpreted as x * y = xy mod 2345 (that is, a multiplication followed by a division with remainder) leads to only 27 multiplications and divisions of integers, which may all be stored in a single machine word.

Signed-digit recoding[edit]

In certain computations it may be more efficient to allow negative coefficients and hence use the inverse of the base, provided inversion in G is "fast" or has been precomputed. For example, when computing x2k−1, the binary method requires k−1 multiplications and k−1 squarings. However, one could perform k squarings to get x2k and then multiply by x−1 to obtain x2k−1.

To this end we define the signed-digit representation of an integer n in radix b as

Signed binary representation corresponds to the particular choice b = 2 and . It is denoted by . There are several methods for computing this representation. The representation is not unique. For example, take n = 478: two distinct signed-binary representations are given by and , where is used to denote −1. Since the binary method computes a multiplication for every non-zero entry in the base-2 representation of n, we are interested in finding the signed-binary representation with the smallest number of non-zero entries, that is, the one with minimal Hamming weight. One method of doing this is to compute the representation in non-adjacent form, or NAF for short, which is one that satisfies and denoted by . For example, the NAF representation of 478 is . This representation always has minimal Hamming weight. A simple algorithm to compute the NAF representation of a given integer with is the following:


for i = 0 to l − 1 do
  
  
return 

Another algorithm by Koyama and Tsuruoka does not require the condition that ; it still minimizes the Hamming weight.

Alternatives and generalizations[edit]

Exponentiation by squaring can be viewed as a suboptimal addition-chain exponentiation algorithm: it computes the exponent by an addition chain consisting of repeated exponent doublings (squarings) and/or incrementing exponents by one (multiplying by x) only. More generally, if one allows any previously computed exponents to be summed (by multiplying those powers of x), one can sometimes perform the exponentiation using fewer multiplications (but typically using more memory). The smallest power where this occurs is for n = 15:

 (squaring, 6 multiplies),
 (optimal addition chain, 5 multiplies if x3 is re-used).

In general, finding the optimal addition chain for a given exponent is a hard problem, for which no efficient algorithms are known, so optimal chains are typically only used for small exponents (e.g. in compilers where the chains for small powers have been pre-tabulated). However, there are a number of heuristic algorithms that, while not being optimal, have fewer multiplications than exponentiation by squaring at the cost of additional bookkeeping work and memory usage. Regardless, the number of multiplications never grows more slowly than Θ(log n), so these algorithms only improve asymptotically upon exponentiation by squaring by a constant factor at best.

See also[edit]

Notes[edit]

  1. ^ In this line, the loop finds the longest string of length less than or equal to k ending in a non-zero value. Not all odd powers of 2 up to need be computed, and only those specifically involved in the computation need be considered.

References[edit]

  1. ^ a b Cohen, H.; Frey, G., eds. (2006). Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and Its Applications. Chapman & Hall/CRC. ISBN 9781584885184.
  2. ^ Montgomery, Peter L. (1987). "Speeding the Pollard and Elliptic Curve Methods of Factorization" (PDF). Math. Comput. 48 (177): 243–264. doi:10.1090/S0025-5718-1987-0866113-7.
  3. ^ Gueron, Shay (5 April 2012). "Efficient software implementations of modular exponentiation" (PDF). Journal of Cryptographic Engineering. 2 (1): 31–43. doi:10.1007/s13389-012-0031-5.