Modular Multiplicative Inverse

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Let's get this over with.

Modular Multiplicative Inverse

In the intricate tapestry of mathematics, specifically within the domain of arithmetic, the concept of a modular multiplicative inverse is as crucial as a perfectly timed silence in a room full of chatter. For an integer, let's call it 'a', its modular multiplicative inverse, denoted by 'x', is an integer that, when multiplied by 'a', yields a result congruent to 1 with respect to a given modulus 'm'. This isn't just some abstract notion; it's a fundamental building block for solving certain types of equations.

In the precise language of modular arithmetic, this relationship is formally expressed as:

$ax \equiv 1 \pmod{m}$

This notation is a concise way of stating that 'm' divides the difference ' $ax - 1$ ' evenly. In simpler terms, when you divide the product ' $ax$ ' by 'm', the remainder is precisely 1. It’s like finding a key that unlocks a specific lock, but the lock is a number and the key is another number that, when combined, return you to a base state of '1'.

Now, if an integer 'a' possesses a modular multiplicative inverse modulo 'm', it's not a solitary affair. There exists an infinite set of solutions to this congruence, all belonging to the same congruence class with respect to the modulus 'm'. Furthermore, any integer that shares the same congruence class as 'a' will have an inverse within the same congruence class as 'x'. This can be elegantly represented using the notation $\overline{w}$ to denote the congruence class containing 'w'. Thus, the modular multiplicative inverse of the congruence class $\overline{a}$ is the congruence class $\overline{x}$ such that:

$\overline{a} \cdot _{m} \overline{x} = \overline{1}$

Here, $\cdot _{m}$ signifies the multiplication of equivalence classes modulo 'm'. This representation strikingly mirrors the concept of a multiplicative inverse in the familiar realms of rational or real numbers, substituting individual numbers with their equivalence classes and adapting the binary operation accordingly.

The utility of this operation is profound, particularly in solving linear congruences of the form:

$ax \equiv b \pmod{m}$

When the modular multiplicative inverse of 'a' is known, solving for 'x' becomes a much more direct process. Beyond theoretical elegance, modular multiplicative inverses play a vital role in practical applications, most notably in cryptography, including public-key cryptography and the widely recognized RSA algorithm. The efficiency of finding these inverses is bolstered by the existence of a remarkably swift algorithm: the extended Euclidean algorithm.

Modular Arithmetic

At its core, modular arithmetic deals with remainders. For any given positive integer 'm', known as the modulus, two integers, 'a' and 'b', are considered congruent modulo 'm' if their difference ('a - b') is perfectly divisible by 'm'. This relationship is denoted as:

$a \equiv b \pmod{m}$

This congruence defines an equivalence relation over the set of all integers, $\mathbb{Z}$ . The resulting equivalence classes, partitioning the integers, are termed congruence classes modulo 'm', or sometimes residue classes modulo 'm'. If we use $\overline{a}$ to represent the congruence class containing integer 'a', then:

$\overline{a} = \{b \in \mathbb{Z} \mid a \equiv b \pmod{m}\}$

This means $\overline{a}$ is the set of all integers that leave the same remainder as 'a' when divided by 'm'.

A linear congruence, such as $ax \equiv b \pmod{m}$ , is a fundamental equation within this system. Unlike their counterparts in the realm of real numbers, linear congruences don't always behave predictably. They can have zero solutions, precisely one solution, or a multitude of solutions. Crucially, if 'x' is a solution, then any integer belonging to the same congruence class as 'x' ( $\overline{x}$ ) is also a solution. Therefore, when we discuss the number of solutions, we are referring to the distinct congruence classes that contain solutions.

The existence of solutions hinges on a simple condition: if 'd' is the greatest common divisor of 'a' and 'm' (i.e., $d = \gcd(a, m)$ ), then the congruence $ax \equiv b \pmod{m}$ has solutions if and only if 'd' divides 'b'. If this condition is met, there will be exactly 'd' distinct solutions, or 'd' distinct congruence classes containing solutions.

The Modular Multiplicative Inverse: A Deeper Dive

A modular multiplicative inverse of an integer 'a' with respect to modulus 'm' is, in essence, a specific solution to the linear congruence:

$ax \equiv 1 \pmod{m}$

As established, a solution exists if and only if $\gcd(a, m) = 1$ . This means 'a' and 'm' must be relatively prime, or coprime. When this condition is satisfied, the inverse is not just guaranteed to exist, but it is also unique. If both 'b' and 'b'' were modular multiplicative inverses of 'a' modulo 'm', then:

$ab \equiv ab' \equiv 1 \pmod{m}$

This implies:

$a(b - b') \equiv 0 \pmod{m}$

If 'a' were congruent to 0 modulo 'm', then $\gcd(a, m) = m$ , and 'a' would not have a modular multiplicative inverse. Therefore, it must be the case that $b \equiv b' \pmod{m}$ .

The notation $x \equiv a^{-1} \pmod{m}$ is often used to represent the modular multiplicative inverse. However, this can be a source of confusion, as it might be mistaken for the standard reciprocal of 'a', which is generally not an integer unless 'a' is 1 or -1. This notation is more accurately interpreted when 'a' is understood as a representative of its congruence class $\overline{a}$ , and $a^{-1}$ represents the inverse congruence class.

Integers Modulo m: Building Blocks of Structure

The congruence relation modulo 'm' elegantly partitions the infinite set of integers into exactly 'm' distinct congruence classes. Within this framework, we can define operations of addition and multiplication on these classes. To add or multiply two congruence classes, we simply select a representative integer from each class, perform the standard integer operation (addition or multiplication) on these representatives, and then identify the congruence class that the result belongs to. Symbolically, using $+_{m}$ and $\cdot _{m}$ for operations on congruence classes:

$\overline{a} +_{m} \overline{b} = \overline{a+b}$ $\overline{a} \cdot _{m} \overline{b} = \overline{ab}$

These operations are well-defined, meaning the outcome is independent of the specific representatives chosen from each class.

These 'm' congruence classes, equipped with these defined operations, form a mathematical structure known as a ring, specifically the ring of integers modulo 'm'. Common notations for this ring include $\mathbb{Z}/m\mathbb{Z}$ or $\mathbb{Z}/m$ . A more simplified notation, $\mathbb{Z}_{m}$ , is often used when the context makes it clear that we're dealing with this specific structure and not some other abstract object.

Historically, these congruence classes were referred to as residue classes modulo 'm', a name derived from the fact that all elements within a class share the same remainder when divided by 'm'. A complete system of residues modulo m is any set of 'm' integers, each belonging to a different congruence class. The division algorithm guarantees that the set $\{0, 1, 2, ..., m-1\}$ forms a complete system of residues, often called the least residue system modulo m. The choice between working with individual integers within a residue system or with the abstract congruence classes of the ring $\mathbb{Z}/m\mathbb{Z}$ often depends on the specific problem at hand.

The Multiplicative Group of Integers Modulo m

Not every element within a complete residue system modulo 'm' possesses a modular multiplicative inverse. Zero, for instance, is never invertible. However, if we remove all elements that are not relatively prime to 'm', the remaining elements form a reduced residue system. Every element in this reduced system is guaranteed to have a modular multiplicative inverse. The number of elements in such a system is given by $\phi(m)$ , where $\phi$ is the Euler totient function – essentially counting the positive integers less than 'm' that are coprime to 'm'.

In any ring with unity, not all elements have a multiplicative inverse; those that do are called units. The set of units within a ring forms a group, known as the group of units of the ring, often denoted as $R^\times$ . For the ring of integers modulo 'm', this group is called the multiplicative group of integers modulo 'm', and it is isomorphic to a reduced residue system. Its size, or order, is precisely $\phi(m)$ .

When 'm' is a prime, say 'p', then $\phi(p) = p-1$ . In this scenario, all non-zero elements of $\mathbb{Z}/p\mathbb{Z}$ possess multiplicative inverses, transforming $\mathbb{Z}/p\mathbb{Z}$ into a finite field. The multiplicative group modulo 'p' then becomes a cyclic group of order $p-1$ .

Example of Structure

Consider an integer $n > 1$ . It can be shown that $n^2 - n + 1$ is the modular multiplicative inverse of $n+1$ with respect to the modulus $n^2$ . This is because $(n+1)(n^2 - n + 1) = n^3 + 1$ . This identity manifests in specific examples: $3 \times 3 \equiv 1 \pmod{4}$ , $4 \times 7 \equiv 1 \pmod{9}$ , and $5 \times 13 \equiv 1 \pmod{16}$ .

Let's examine the modulus 10. Two integers are congruent modulo 10 if their difference is a multiple of 10. For instance, $32 \equiv 2 \pmod{10}$ because $32 - 2 = 30$ , which is divisible by 10. Similarly, $111 \equiv 1 \pmod{10}$ because $111 - 1 = 110$ , also divisible by 10.

Among the ten congruence classes modulo 10, we can identify:

$\overline{0} = \{\dots, -20, -10, 0, 10, 20, \dots\}$ $\overline{1} = \{\dots, -19, -9, 1, 11, 21, \dots\}$ $\overline{5} = \{\dots, -15, -5, 5, 15, 25, \dots\}$ $\overline{9} = \{\dots, -11, -1, 9, 19, 29, \dots\}$

Consider the linear congruence $4x \equiv 5 \pmod{10}$ . This has no solutions. The integers congruent to 5 modulo 10 are all odd, while $4x$ is always even. However, $4x \equiv 6 \pmod{10}$ does have solutions: $x=4$ and $x=9$ . This aligns with the rule: $\gcd(4, 10) = 2$ . Since 2 does not divide 5 but does divide 6, the first congruence has no solutions, while the second has exactly $\gcd(4, 10) = 2$ solutions.

Since $\gcd(3, 10) = 1$ , the congruence $3x \equiv 1 \pmod{10}$ must have a unique solution class. Indeed, $x=7$ satisfies this, as $3 \times 7 = 21$ , and $21 - 1 = 20$ , which is divisible by 10. Other integers like 17 and -3 also satisfy this congruence ( $3 \times 17 - 1 = 50$ , $3 \times (-3) - 1 = -10$ ). All these solutions belong to the congruence class $\overline{7}$ , as $3(7 + 10r) - 1 = 21 + 30r - 1 = 20 + 30r = 10(2 + 3r)$ , which is always divisible by 10.

The product of congruence classes $\overline{5}$ and $\overline{8}$ can be found by taking representatives, say 25 from $\overline{5}$ and -2 from $\overline{8}$ . Their product is $(25)(-2) = -50$ , which belongs to the congruence class $\overline{0}$ . Thus:

$\overline{5} \cdot _{10} \overline{8} = \overline{0}$

The ten congruence classes, with these defined operations, constitute the ring $\mathbb{Z}/10\mathbb{Z}$ .

A complete residue system modulo 10 could be $\{10, -9, 2, 13, 24, -15, 26, 37, 8, 9\}$ . The standard least residue system is $\{0, 1, 2, \dots, 9\}$ . A reduced residue system modulo 10 might be $\{1, 3, 7, 9\}$ . The product of any two congruence classes represented by these numbers is another class within this set. These four classes form a group under multiplication, specifically a cyclic group of order four, generated by either 3 or 7. These are precisely the congruence classes that possess modular multiplicative inverses.

Computation of Modular Inverses

The Extended Euclidean Algorithm

The Euclidean algorithm is the standard method for finding the greatest common divisor (gcd) of two integers, say 'a' and 'm'. For a modular inverse of 'a' modulo 'm' to exist, their gcd must be 1. The extended version of this algorithm doesn't just find the gcd; it also finds integers 'x' and 'y' that satisfy Bézout's identity:

$ax + my = \gcd(a, m)$

If $\gcd(a, m) = 1$ , then the equation becomes $ax + my = 1$ . Rearranging this yields $ax - 1 = (-y)m$ , which directly implies:

$ax \equiv 1 \pmod{m}$

Thus, the 'x' found by the extended Euclidean algorithm is the modular multiplicative inverse of 'a' modulo 'm'. This algorithm is remarkably efficient, running in time $O(\log^2(m))$ (assuming $|a| < m$ ), making it the preferred method for calculating modular inverses.

Utilizing Euler's Theorem

An alternative method for finding modular inverses relies on Euler's theorem. The theorem states that if 'a' is coprime to 'm' ( $\gcd(a, m) = 1$ ), then:

$a^{\phi(m)} \equiv 1 \pmod{m}$

where $\phi$ is the Euler totient function. Since 'a' belongs to the multiplicative group $(\mathbb{Z}/m\mathbb{Z})^\times$ if and only if it's coprime to 'm', we can derive the inverse directly:

$a^{\phi(m)-1} \equiv a^{-1} \pmod{m}$

For the special case where 'm' is a prime 'p', $\phi(p) = p-1$ , and the inverse is given by:

$a^{-1} \equiv a^{p-2} \pmod{p}$

While conceptually straightforward, this method is generally less efficient than the extended Euclidean algorithm. Its primary drawbacks include:

Calculating $\phi(m)$ : The most efficient known way to compute $\phi(m)$ requires the prime factorization of 'm', which is a computationally hard problem. Although, if the factorization is known, calculating $\phi(m)$ is simple.
Exponentiation Cost: Even with efficient modular exponentiation techniques like binary exponentiation or Montgomery reduction, repeated multiplications modulo 'm' can be computationally intensive for large 'm'.

However, a significant advantage of using Euler's theorem is that it avoids conditional branches based on the value of 'a'. This property is crucial in public-key cryptography, where protecting secret values from side-channel attacks is paramount. For this reason, implementations like Curve25519 utilize this approach for inverse computation.

Handling Multiple Inverses

It's possible to compute the inverses of multiple numbers ( $a_i$ ) modulo a common modulus 'm' more efficiently than computing each inverse individually. This can be achieved with a single application of the Euclidean algorithm and a series of multiplications. The core idea involves computing the product of all $a_i$ , inverting that product, and then using this inverse to derive each individual inverse.

A more specific algorithm involves:

Calculating prefix products: $b_i = \prod_{j=1}^{i} a_j = a_i b_{i-1}$ for all $i \le n$ .
Computing the inverse of the final product, $b_n^{-1}$ , using a standard method.
Iteratively computing the individual inverses: For $i$ from $n$ down to 2, calculate $a_i^{-1} = b_i^{-1} b_{i-1}$ and update $b_{i-1}^{-1} = b_i^{-1} a_i$ . Finally, $a_1^{-1} = b_1^{-1}$ .

These multiplications can be structured in a tree-like fashion to leverage parallel computing.

Inverses Modulo Prime Powers

For moduli of the form $p^k$ , where 'p' is a prime number and 'k' is a positive integer, modular multiplicative inverses can be computed very efficiently using Newton-Raphson iteration. This method requires $O(\log k)$ multiplications. The principle is that if $ax \equiv 1 \pmod{p^k}$ , then $ax(2-ax) \equiv 1 \pmod{p^{2k}}$ . This allows us to compute an inverse modulo 'p' or a small power of 'p', and then iteratively refine it to obtain inverses modulo progressively larger prime powers $p^{2^n}$ .

A practical application of this is computing inverses modulo powers of 2. All odd integers are their own modular multiplicative inverses modulo 8, as shown by inspection: $1 \times 1 \equiv 1 \pmod{8}$ , $3 \times 3 = 9 \equiv 1 \pmod{8}$ , $5 \times 5 = 25 \equiv 1 \pmod{8}$ , and $7 \times 7 = 49 \equiv 1 \pmod{8}$ . We can then use Newton-Raphson iterations to compute inverses modulo $2^6$ , $2^{12}$ , $2^{24}$ , and so on.

For instance, in C programming with uint64_t types (which operate modulo $2^{64}$ ), an inverse modulo $2^{64}$ can be found with five Newton-Raphson iterations:

#include <stdint.h>
uint64_t modinv64(uint64_t a) {
  uint64_t x = a;
  for (int i = 0; i < 5; i++) x *= 2 - a*x;
  return x;
}

Optimizations can further enhance this routine. Pre-computed lookup tables can provide an initial inverse modulo a larger power of two, and on some systems, 32-bit multiplications might be faster. This leads to routines like the following, which leverages a lookup table for an initial inverse modulo $2^8$ :

#include <stdint.h>
uint64_t modinv64(uint64_t a) {
  static const uint8_t tbl[256] = {
    0,1,0,171,0,205,0,183,0,57,0,163,0,197,0,239,
    0,241,0,27,0,61,0,167,0,41,0,19,0,53,0,223,
    0,225,0,139,0,173,0,151,0,25,0,131,0,165,0,207,
    0,209,0,251,0,29,0,135,0,9,0,243,0,21,0,191,
    0,193,0,107,0,141,0,119,0,249,0,99,0,133,0,175,
    0,177,0,219,0,253,0,103,0,233,0,211,0,245,0,159,
    0,161,0,75,0,109,0,87,0,217,0,67,0,101,0,143,
    0,145,0,187,0,221,0,71,0,201,0,179,0,213,0,127,
    0,129,0,43,0,77,0,55,0,185,0,35,0,69,0,111,
    0,113,0,155,0,189,0,39,0,169,0,147,0,181,0,95,
    0,97,0,11,0,45,0,23,0,153,0,3,0,37,0,79,
    0,81,0,123,0,157,0,7,0,137,0,115,0,149,0,63,
    0,65,0,235,0,13,0,247,0,121,0,227,0,5,0,47,
    0,49,0,91,0,125,0,231,0,105,0,83,0,117,0,31,
    0,33,0,203,0,237,0,215,0,89,0,195,0,229,0,15,
    0,17,0,59,0,93,0,199,0,73,0,51,0,85,0,255 };
  uint32_t a32 = (uint32_t)a;
  uint32_t x = tbl[a & 0xFF]; // inverse modulo 2^8
  x *= 2 - a32*x; // 32-bit multiplications
  x *= 2 - a32*x; // 32-bit multiplications
  return x * (2 - a*x); // 64-bit multiplications
}

Applications

The utility of modular multiplicative inverses permeates numerous algorithms that rely on the principles of modular arithmetic. In cryptography, for instance, their use allows for operations that are both computationally swift and require minimal storage, while simultaneously making other operations more challenging. This duality is exploited to great effect. In the RSA algorithm, the processes of encryption and decryption are performed using a pair of numbers that are multiplicative inverses with respect to a carefully chosen modulus. One of these numbers is publicly known, enabling rapid encryption, while its inverse, kept secret, is used for decryption. The difficulty of deriving the secret inverse from the public number is the bedrock of the system's security and privacy.

Consider another example from computer science: the problem of exact division. Suppose you have a list of odd, word-sized integers, all divisible by some number 'k'. To divide them all by 'k':

Use the extended Euclidean algorithm to compute $k^{-1} \pmod{2^w}$ , where 'w' is the number of bits in a word. This inverse is guaranteed to exist because 'k' is odd and $2^w$ has no odd factors.
For each number in the list, multiply it by $k^{-1}$ and take the least significant word of the result.

On many processors, particularly those lacking hardware division support, division is a significantly slower operation than multiplication. This technique can therefore yield substantial performance gains, as the initial inverse calculation is performed only once.

Modular multiplicative inverses are also instrumental in constructing solutions for systems of linear congruences, as guaranteed by the Chinese Remainder Theorem.

For instance, consider the system:

$X \equiv 4 \pmod{5}$ $X \equiv 4 \pmod{7}$ $X \equiv 6 \pmod{11}$

This system has a common solution because 5, 7, and 11 are pairwise coprime. A solution can be expressed as:

$X = t_1 (7 \times 11) \times 4 + t_2 (5 \times 11) \times 4 + t_3 (5 \times 7) \times 6$

where:

$t_1 = 3$ is the modular multiplicative inverse of $7 \times 11 \pmod{5}$
$t_2 = 6$ is the modular multiplicative inverse of $5 \times 11 \pmod{7}$
$t_3 = 6$ is the modular multiplicative inverse of $5 \times 7 \pmod{11}$

Substituting these values:

$X = 3 \times (7 \times 11) \times 4 + 6 \times (5 \times 11) \times 4 + 6 \times (5 \times 7) \times 6 = 3504$

The unique reduced solution modulo the Least common multiple of 5, 7, and 11 (which is 385) is:

$X \equiv 3504 \equiv 39 \pmod{385}$

Furthermore, the modular multiplicative inverse plays a key role in the definition of the Kloosterman sum.