Montgomery modular multiplication

Template:Short description In modular arithmetic computation, Montgomery modular multiplication, more commonly referred to as Montgomery multiplication, is a method for performing fast modular multiplication. It was introduced in 1985 by the American mathematician Peter L. Montgomery.

Montgomery modular multiplication relies on a special representation of numbers called Montgomery form. The algorithm uses the Montgomery forms of Template:Mvar and Template:Mvar to efficiently compute the Montgomery form of ab mod N. The efficiency comes from avoiding expensive division operations. Classical modular multiplication reduces the double-width product ab using division by Template:Mvar and keeping only the remainder. This division requires quotient digit estimation and correction. The Montgomery form, in contrast, depends on a constant Template:Mvar which is coprime to Template:Mvar, and the only division necessary in Montgomery multiplication is division by Template:Mvar. The constant Template:Mvar can be chosen so that division by Template:Mvar is easy, significantly improving the speed of the algorithm. In binary computers, Template:Mvar is always a power of two, since division by powers of two can be implemented by bit shifting.

The need to convert Template:Mvar and Template:Mvar into Montgomery form and their product out of Montgomery form means that computing a single product by Montgomery multiplication is slower than the conventional or Barrett reduction algorithms. However, when performing many multiplications in a row, as in modular exponentiation, intermediate results can be left in Montgomery form. Then the initial and final conversions become a negligible fraction of the overall computation. Many important cryptosystems such as RSA and Diffie–Hellman key exchange are based on arithmetic operations modulo a large odd number, and for these cryptosystems, computations using Montgomery multiplication with Template:Mvar a power of two are faster than the available alternatives.

Modular arithmetic

Let Template:Mvar denote a positive integer modulus. The quotient ring Z/NZ consists of residue classes modulo Template:Mvar, that is, its elements are sets of the form

$\{a+kN:k\in \mathbf{𝐙}\},$

where Template:Mvar ranges across the integers. Each residue class is a set of integers such that the difference of any two integers in the set is divisible by Template:Mvar (and the residue class is maximal with respect to that property; integers aren't left out of the residue class unless they would violate the divisibility condition). The residue class corresponding to Template:Mvar is denoted a. Equality of residue classes is called congruence and is denoted

$\overline{a}\equiv \overline{b}(\mathrm{mod}N).$

Storing an entire residue class on a computer is impossible because the residue class has infinitely many elements. Instead, residue classes are stored as representatives. Conventionally, these representatives are the integers Template:Mvar for which 0 ≤ a ≤ N − 1. If Template:Mvar is an integer, then the representative of a is written a mod N. When writing congruences, it is common to identify an integer with the residue class it represents. With this convention, the above equality is written a ≡ b mod N.

Arithmetic on residue classes is done by first performing integer arithmetic on their representatives. The output of the integer operation determines a residue class, and the output of the modular operation is determined by computing the residue class's representative. For example, if N = 17, then the sum of the residue classes 7 and 15 is computed by finding the integer sum 7 + 15 = 22, then determining 22 mod 17, the integer between 0 and 16 whose difference with 22 is a multiple of 17. In this case, that integer is 5, so 7 + 15 ≡ 5 mod 17.

Montgomery form

If Template:Mvar and Template:Mvar are integers in the range [0, N − 1], then their sum is in the range [0, 2N − 2] and their difference is in the range [−N + 1, N − 1], so determining the representative in [0, N − 1] requires at most one subtraction or addition (respectively) of Template:Mvar. However, the product ab is in the range [0, N² − 2N + 1]. Storing the intermediate integer product ab requires twice as many bits as either Template:Mvar or Template:Mvar, and efficiently determining the representative in [0, N − 1] requires division. Mathematically, the integer between 0 and N − 1 that is congruent to ab can be expressed by applying the Euclidean division theorem:

$ab=qN+r,$

where Template:Mvar is the quotient $\lfloor ab/N\rfloor$ and Template:Mvar, the remainder, is in the interval [0, N − 1]. The remainder Template:Mvar is ab mod N. Determining Template:Mvar can be done by computing Template:Mvar, then subtracting qN from ab. For example, again with $N=17$, the product 7 ⋅ 15 is determined by computing $7\cdot 15=105$, dividing $\lfloor 105/17\rfloor =6$, and subtracting $105-6\cdot 17=105-102=3$.

Because the computation of Template:Mvar requires division, it is undesirably expensive on most computer hardware. Montgomery form is a different way of expressing the elements of the ring in which modular products can be computed without expensive divisions. While divisions are still necessary, they can be done with respect to a different divisor Template:Mvar. This divisor can be chosen to be a power of two, for which division can be replaced by shifting, or a whole number of machine words, for which division can be replaced by omitting words. These divisions are fast, so most of the cost of computing modular products using Montgomery form is the cost of computing ordinary products.

The auxiliary modulus Template:Mvar must be a positive integer such that gcd(R, N) = 1. For computational purposes it is also necessary that division and reduction modulo Template:Mvar are inexpensive, and the modulus is not useful for modular multiplication unless R > N. The Montgomery form of the residue class a with respect to Template:Mvar is aR mod N, that is, it is the representative of the residue class aR. For example, suppose that N = 17 and that R = 100. The Montgomery forms of 3, 5, 7, and 15 are 300 mod 17 = 11, 500 mod 17 = 7, 700 mod 17 = 3, and 1500 mod 17 = 4.

Addition and subtraction in Montgomery form are the same as ordinary modular addition and subtraction because of the distributive law:

$aR+bR=(a+b)R,$

$aR-bR=(a-b)R.$

Note that doing the operation in Montgomery form does not lose information compared to doing it in the quotient ring Z/NZ. This is a consequence of the fact that, because gcd(R, N) = 1, multiplication by Template:Mvar is an isomorphism on the additive group Z/NZ. For example, (7 + 15) mod 17 = 5, which in Montgomery form becomes (3 + 4) mod 17 = 7.

Multiplication in Montgomery form, however, is seemingly more complicated. The usual product of aR and bR does not represent the product of Template:Mvar and Template:Mvar because it has an extra factor of Template:Mvar:

$(aRmodN)(bRmodN)modN=(abR)RmodN.$

Computing products in Montgomery form requires removing the extra factor of Template:Mvar. While division by Template:Mvar is cheap, the intermediate product (aR mod N)(bR mod N) is not divisible by Template:Mvar because the modulo operation has destroyed that property. So for instance, the product of the Montgomery forms of 7 and 15 modulo 17, with R = 100, is the product of 3 and 4, which is 12. Since 12 is not divisible by 100, additional effort is required to remove the extra factor of Template:Mvar.

Removing the extra factor of Template:Mvar can be done by multiplying by an integer R′ such that RR′ ≡ 1 (mod N), that is, by an R′ whose residue class is the modular inverse of Template:Mvar mod Template:Mvar. Then, working modulo Template:Mvar,

$(aRmodN)(bRmodN)R^{\prime }\equiv (aR)(bR)R^{-1}\equiv (ab)R(\mathrm{mod}N).$

The integer R′ exists because of the assumption that Template:Mvar and Template:Mvar are coprime. It can be constructed using the extended Euclidean algorithm. The extended Euclidean algorithm efficiently determines integers R′ and N′ that satisfy Bézout's identity: 0 < R′ < N, 0 < N′ < R, and:

$R{R}^{\prime }-N{N}^{\prime }=1.$

This shows that it is possible to do multiplication in Montgomery form. A straightforward algorithm to multiply numbers in Montgomery form is therefore to multiply aR mod N, bR mod N, and R′ as integers and reduce modulo Template:Mvar.

For example, to multiply 7 and 15 modulo 17 in Montgomery form, again with R = 100, compute the product of 3 and 4 to get 12 as above. The extended Euclidean algorithm implies that 8⋅100 − 47⋅17 = 1, so R′ = 8. Multiply 12 by 8 to get 96 and reduce modulo 17 to get 11. This is the Montgomery form of 3, as expected.

The REDC algorithm

While the above algorithm is correct, it is slower than multiplication in the standard representation because of the need to multiply by R′ and divide by Template:Mvar. Montgomery reduction, also known as REDC, is an algorithm that simultaneously computes the product by R′ and reduces modulo Template:Mvar more quickly than the naïve method. Unlike conventional modular reduction, which focuses on making the number smaller than Template:Mvar, Montgomery reduction focuses on making the number more divisible by Template:Mvar. It does this by adding a small multiple of Template:Mvar which is sophisticatedly chosen to cancel the residue modulo Template:Mvar. Dividing the result by Template:Mvar yields a much smaller number. This number is so much smaller that it is nearly the reduction modulo Template:Mvar, and computing the reduction modulo Template:Mvar requires only a final conditional subtraction. Because all computations are done using only reduction and divisions with respect to Template:Mvar, not Template:Mvar, the algorithm runs faster than a straightforward modular reduction by division.

function REDC is
    input: Integers R and N with gcd(R, N) = 1,
           Integer N′ in [0, R − 1] such that NN′ ≡ −1 mod R,
           Integer T in the range [0, RN − 1].
    output: Integer S in the range [0, N − 1] such that S ≡ TR−1 mod N

    m ← ((T mod R)N′) mod R
    t ← (T + mN) / R
    if t ≥ N then
        return t − N
    else
        return t
    end if
end function

To see that this algorithm is correct, first observe that Template:Mvar is chosen precisely so that T + mN is divisible by Template:Mvar. A number is divisible by Template:Mvar if and only if it is congruent to zero mod Template:Mvar, and we have:

$T+mN\equiv T+(((TmodR)N^{\prime })modR)N\equiv T+T{N}^{\prime }N\equiv T-T\equiv 0(\mathrm{mod}R).$

Therefore, Template:Mvar is an integer. Second, the output is either Template:Mvar or t − N, both of which are congruent to t mod N, so to prove that the output is congruent to TR⁻¹ mod N, it suffices to prove that Template:Mvar is TR⁻¹ mod N, Template:Mvar satisfies:

$t\equiv (T+mN)R^{-1}\equiv T{R}^{-1}+(m{R}^{-1})N\equiv T{R}^{-1}(\mathrm{mod}N).$

Therefore, the output has the correct residue class. Third, Template:Mvar is in [0, R − 1], and therefore T + mN is between 0 and (RN − 1) + (R − 1)N < 2RN. Hence Template:Mvar is less than 2N, and because it's an integer, this puts Template:Mvar in the range [0, 2N − 1]. Therefore, reducing Template:Mvar into the desired range requires at most a single subtraction, so the algorithm's output lies in the correct range.

To use REDC to compute the product of 7 and 15 modulo 17, first convert to Montgomery form and multiply as integers to get 12 as above. Then apply REDC with R = 100, N = 17, N′ = 47, and T = 12. The first step sets Template:Mvar to 12 ⋅ 47 mod 100 = 64. The second step sets Template:Mvar to (12 + 64 ⋅ 17) / 100. Notice that 12 + 64 ⋅ 17 is 1100, a multiple of 100 as expected. Template:Mvar is set to 11, which is less than 17, so the final result is 11, which agrees with the computation of the previous section.

As another example, consider the product 7 ⋅ 15 mod 17 but with R = 10. Using the extended Euclidean algorithm, compute −5 ⋅ 10 + 3 ⋅ 17 = 1, so N′ will be −3 mod 10 = 7. The Montgomery forms of 7 and 15 are 70 mod 17 = 2 and 150 mod 17 = 14, respectively. Their product 28 is the input Template:Mvar to REDC, and since 28 < RN = 170, the assumptions of REDC are satisfied. To run REDC, set Template:Mvar to (28 mod 10) ⋅ 7 mod 10 = 196 mod 10 = 6. Then 28 + 6 ⋅ 17 = 130, so t = 13. Because 30 mod 17 = 13, this is the Montgomery form of 3 = 7 ⋅ 15 mod 17.

Interpretation via the Chinese Remainder Theorem

Source:

Given the modulus Template:Mvar and the Montgomery radix Template:Mvar used in a Montgomery reduction, consider the residue ring

$\mathbb{\mathbb{Z}}/(NR)\mathbb{\mathbb{Z}}\cong \mathbb{\mathbb{Z}}/N\mathbb{\mathbb{Z}}\times \mathbb{\mathbb{Z}}/R\mathbb{\mathbb{Z}},$

an isomorphism that follows from the Chinese Remainder Theorem (CRT).

CRT reconstruction for an intermediate product

For an integer $T$ with $0\le T<NR$ (as is typical when $T$ arises from multiplying two residues), take its reductions

$T_N=TmodN,T_R=TmodR.$

The CRT gives the explicit reconstruction formula

$T\equiv T_N\left(R^{-1}modN\right)R+T_R\left(N^{-1}modR\right)N(\mathrm{mod}NR).$

Because the right-hand side is already taken modulo $NR$, this may also be written as

$T\equiv \left(T_N{R}^{-1}modN\right)R+\left(T_R{N}^{-1}modR\right)N(\mathrm{mod}NR).$

Both summands lie in the half‑open interval $[0,NR)$:

$0\le \left(T_N{R}^{-1}modN\right)R<NR,0\le \left(T_R{N}^{-1}modR\right)N<NR.$

Hence, as integer equations (not merely congruences) we have

$T=\left(T_N{R}^{-1}modN\right)R+\left(T_R{N}^{-1}modR\right)N,$

or,

$T+NR=\left(T_N{R}^{-1}modN\right)R+\left(T_R{N}^{-1}modR\right)N.$

Isolating the term containing T mod N

To solve for $T_N$, isolate the first summand:

$\left(T_N{R}^{-1}modN\right)R=\{\begin{array}{l}T-\left(T_R{N}^{-1}modR\right)N,\\ T+NR-\left(T_R{N}^{-1}modR\right)N.\end{array}$

Every quantity above is an integer, and the left‑hand side is a multiple of $R$; therefore each right‑hand side is divisible by $R$. Dividing by $R$ yields

$T_N{R}^{-1}modN=\{\begin{array}{l}\frac{T-\left(T_R{N}^{-1}modR\right)N}{R},\\ \frac{T+NR-\left(T_R{N}^{-1}modR\right)N}{R}=\frac{T-\left(T_R{N}^{-1}modR\right)N}{R}+N.\end{array}$

Resulting relations

Consequently,

$\frac{T-\left(T_R{N}^{-1}modR\right)N}{R}=\{\begin{array}{l}{T}_N{R}^{-1}modN,\\ T_N{R}^{-1}modN+N.\end{array}$

This gives two key facts:

• Congruence

$\frac{T-\left(T_R{N}^{-1}modR\right)N}{R}\equiv T_N{R}^{-1}(\mathrm{mod}N).$

• Numeric bound

$0\le \frac{T-\left(T_R{N}^{-1}modR\right)N}{R}<2N.$

Therefore, by reducing

$\frac{T-\left(T_R{N}^{-1}modR\right)N}{R}$

once more modulo Template:Mvar, one obtains the non‑negative residue representing $T_N{R}^{-1}modN$.

Arithmetic in Montgomery form

Many operations of interest modulo Template:Mvar can be expressed equally well in Montgomery form. Addition, subtraction, negation, comparison for equality, multiplication by an integer not in Montgomery form, and greatest common divisors with Template:Mvar may all be done with the standard algorithms. The Jacobi symbol can be calculated as $\left(\frac{a}{N}\right)=\left(\frac{aR}{N}\right)/\left(\frac{R}{N}\right)$ as long as $\left(\frac{R}{N}\right)$ is stored.

When R > N, most other arithmetic operations can be expressed in terms of REDC. This assumption implies that the product of two representatives mod Template:Mvar is less than Template:Mvar, the exact hypothesis necessary for REDC to generate correct output. In particular, the product of aR mod N and bR mod N is REDC((aR mod N)(bR mod N)). The combined operation of multiplication and REDC is often called Montgomery multiplication.

Conversion into Montgomery form is done by computing REDC((a mod N)(R² mod N)). Conversion out of Montgomery form is done by computing REDC(aR mod N). The modular inverse of aR mod N is REDC((aR mod N)⁻¹(R³ mod N)). Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, that is, to R mod N, and by replacing the multiply and square steps by Montgomery multiplies.

Performing these operations requires knowing at least N′ and R² mod N. When Template:Mvar is a power of a small positive integer Template:Mvar, N′ can be computed by Hensel's lemma: The inverse of Template:Mvar modulo Template:Mvar is computed by a naïve algorithm (for instance, if b = 2 then the inverse is 1), and Hensel's lemma is used repeatedly to find the inverse modulo higher and higher powers of Template:Mvar, stopping when the inverse modulo Template:Mvar is known; N′ is the negation of this inverse. The constants R mod N and R³ mod N can be generated as REDC(R² mod N) and as REDC((R² mod N)(R² mod N)). The fundamental operation is to compute REDC of a product. When standalone REDC is needed, it can be computed as REDC of a product with 1 mod N. The only place where a direct reduction modulo Template:Mvar is necessary is in the precomputation of R² mod N.

Montgomery arithmetic on multiprecision integers

Most cryptographic applications require numbers that are hundreds or even thousands of bits long. Such numbers are too large to be stored in a single machine word. Typically, the hardware performs multiplication mod some base Template:Mvar, so performing larger multiplications requires combining several small multiplications. The base Template:Mvar is typically 2 for microelectronic applications, 2⁸ for 8-bit firmware, or 2³² or 2⁶⁴ for software applications.

The REDC algorithm requires products modulo Template:Mvar, and typically R > N so that REDC can be used to compute products. However, when Template:Mvar is a power of Template:Mvar, there is a variant of REDC which requires products only of machine word sized integers. Suppose that positive multi-precision integers are stored little endian, that is, Template:Mvar is stored as an array x[0], ..., x[ℓ - 1] such that 0 ≤ x[i] < B for all Template:Mvar and x = ∑ x[i] Bⁱ. The algorithm begins with a multiprecision integer Template:Mvar and reduces it one word at a time. First an appropriate multiple of Template:Mvar is added to make Template:Mvar divisible by Template:Mvar. Then a multiple of Template:Mvar is added to make Template:Mvar divisible by B², and so on. Eventually Template:Mvar is divisible by Template:Mvar, and after division by Template:Mvar the algorithm is in the same place as REDC was after the computation of Template:Mvar.

function MultiPrecisionREDC is
    Input: Integer N with gcd(B, N) = 1, stored as an array of p words,
           Integer R = Br,     --thus, r = logB R
           Integer N′ in [0, B − 1] such that NN′ ≡ −1 (mod B),
           Integer T in the range 0 ≤ T < RN, stored as an array of r + p words.

    Output: Integer S in [0, N − 1] such that TR−1 ≡ S (mod N), stored as an array of p words.

    Set T[r + p] = 0  (extra carry word)
    for 0 ≤ i < r do
        --loop1- Make T divisible by Bi+1

        c ← 0
        m ← T[i] ⋅ N′ mod B
        for 0 ≤ j < p do
            --loop2- Add the m ⋅ N[j] and the carry from earlier, and find the new carry

            x ← T[i + j] + m ⋅ N[j] + c
            T[i + j] ← x mod B
            c ← ⌊x / B⌋
        end for
        for p ≤ j ≤ r + p − i do
            --loop3- Continue carrying

            x ← T[i + j] + c
            T[i + j] ← x mod B
            c ← ⌊x / B⌋
        end for
    end for

    for 0 ≤ i ≤ p do
        S[i] ← T[i + r]
    end for

    if S ≥ N then
        return S − N
    else
        return Template:Var
    end if
end function

The final comparison and subtraction is done by the standard algorithms.

The above algorithm is correct for essentially the same reasons that REDC is correct. Each time through the Template:Mvar loop, Template:Mvar is chosen so that T[i] + mN[0] is divisible by Template:Mvar. Then Template:Mvar is added to Template:Mvar. Because this quantity is zero mod Template:Mvar, adding it does not affect the value of T mod N. If Template:Mvar denotes the value of Template:Mvar computed in the Template:Mvarth iteration of the loop, then the algorithm sets Template:Mvar to T + (∑ m_i Bⁱ)N. Because MultiPrecisionREDC and REDC produce the same output, this sum is the same as the choice of Template:Mvar that the REDC algorithm would make.

The last word of Template:Mvar, T[r + p] (and consequently S[p]), is used only to hold a carry, as the initial reduction result is bound to a result in the range of 0 ≤ S < 2N. It follows that this extra carry word can be avoided completely if it is known in advance that R ≥ 2N. On a typical binary implementation, this is equivalent to saying that this carry word can be avoided if the number of bits of Template:Mvar is smaller than the number of bits of Template:Mvar. Otherwise, the carry will be either zero or one. Depending upon the processor, it may be possible to store this word as a carry flag instead of a full-sized word.

It is possible to combine multiprecision multiplication and REDC into a single algorithm. This combined algorithm is usually called Montgomery multiplication. Several different implementations are described by Koç, Acar, and Kaliski. The algorithm may use as little as p + 2 words of storage (plus a carry bit).

As an example, let B = 10, N = 997, and R = 1000. Suppose that a = 314 and b = 271. The Montgomery representations of Template:Mvar and Template:Mvar are 314000 mod 997 = 942 and 271000 mod 997 = 813. Compute 942 ⋅ 813 = 765846. The initial input Template:Mvar to MultiPrecisionREDC will be [6, 4, 8, 5, 6, 7]. The number Template:Mvar will be represented as [7, 9, 9]. The extended Euclidean algorithm says that −299 ⋅ 10 + 3 ⋅ 997 = 1, so N′ will be 7.

i ← 0
m ← 6 ⋅ 7 mod 10 = 2

j T       c
- ------- -
0 0485670 2    (After first iteration of first loop)
1 0485670 2
2 0485670 2
3 0487670 0    (After first iteration of second loop)
4 0487670 0
5 0487670 0
6 0487670 0

i ← 1
m ← 4 ⋅ 7 mod 10 = 8

j T       c
- ------- -
0 0087670 6    (After first iteration of first loop)
1 0067670 8
2 0067670 8
3 0067470 1    (After first iteration of second loop)
4 0067480 0
5 0067480 0

i ← 2
m ← 6 ⋅ 7 mod 10 = 2

j T       c
- ------- -
0 0007480 2    (After first iteration of first loop)
1 0007480 2
2 0007480 2
3 0007400 1    (After first iteration of second loop)
4 0007401 0

Therefore, before the final comparison and subtraction, S = 1047. The final subtraction yields the number 50. Since the Montgomery representation of 314 ⋅ 271 mod 997 = 349 is 349000 mod 997 = 50, this is the expected result.

When working in base 2, determining the correct Template:Mvar at each stage is particularly easy: If the current working bit is even, then Template:Mvar is zero and if it's odd, then Template:Mvar is one. Furthermore, because each step of MultiPrecisionREDC requires knowing only the lowest bit, Montgomery multiplication can be easily combined with a carry-save adder.

Side-channel attacks

Because Montgomery reduction avoids the correction steps required in conventional division when quotient digit estimates are inaccurate, it is mostly free of the conditional branches which are the primary targets of timing and power side-channel attacks; the sequence of instructions executed is independent of the input operand values. The only exception is the final conditional subtraction of the modulus, but it is easily modified (to always subtract something, either the modulus or zero) to make it resistant. It is of course necessary to ensure that the exponentiation algorithm built around the multiplication primitive is also resistant.Template:R

See also

• Barrett reduction

External links

• Template:Cite CiteSeerX