Lib.rs

bitcoinsecp256k1-modinv

This Rust crate is part of the Bitcoin system and is a direct translation from the C++ codebase. The primary purpose of this crate is to provide efficient modular inversion operations for the finite field used in the secp256k1 elliptic curve. The functions provided in this crate are optimized for 64-bit and 32-bit architectures, with the ModInv64 and ModInv32 structs respectively.

The modinv operation is used to calculate the modular inverse of a number a with respect to a modulus m. That is, to find a number b such that a * b ≡ 1 (mod m). The modular inverse is an important operation in public-key cryptography, where it is used to compute private keys from public keys, and to verify digital signatures.

The crate provides various implementations of the modinv operation, optimized for different architectures. The algorithms implemented include the classical binary extended Euclidean algorithm, as well as more efficient variants such as Montgomery's ladder and sliding window algorithms.

In addition to the modinv operation, the crate also provides other modular arithmetic functions such as multiplication and normalization. These functions are used internally by the modular inversion algorithms.

Note that since modular inversion is a computationally intensive operation, it is important to use efficient implementations in order to achieve acceptable performance in cryptographic applications. The functions provided by this crate are optimized for speed and should be used whenever modular inversion is required in a secp256k1-based system.

Notice: This crate is part of a direct translation from C++ to Rust of the bitcoin core. As such, some of the function bodies may still be in the process of translation. Please note that this system will become testable once the translation process is complete.

163> could you please explain the classical binary extended euclidean algorithm? also please explain montgomery's ladder and the sliding window algorithms

Sure, here's an explanation of the classical binary extended Euclidean algorithm, Montgomery's ladder algorithm, and the sliding window algorithm.

Classical Binary Extended Euclidean Algorithm:

The extended Euclidean algorithm is used to compute the greatest common divisor (GCD) of two integers, along with two coefficients that can be used to express the GCD as a linear combination of the two integers. The binary extended Euclidean algorithm is an optimized version of the extended Euclidean algorithm that operates on binary inputs, which is particularly useful in cryptography.

In the binary extended Euclidean algorithm, we represent the input integers as binary numbers and apply a series of binary operations to compute the GCD and coefficients. Specifically, we repeatedly divide the larger number by 2 until both numbers become odd, and then apply a series of binary subtraction and addition operations to compute the GCD and coefficients. This algorithm is more efficient than the classical extended Euclidean algorithm because it operates on binary inputs, which allows us to perform binary shifts and bit-wise operations instead of division and multiplication.

Montgomery's ladder algorithm:

Montgomery's ladder algorithm is a scalar multiplication algorithm that is used to compute the scalar multiplication of a point on an elliptic curve. Scalar multiplication is the operation of multiplying a point by a scalar (a non-negative integer) in order to generate a new point on the curve.

The Montgomery ladder algorithm is particularly useful in elliptic curve cryptography because it is resistant to side-channel attacks. These attacks are a type of attack that exploit information leaked by the system (such as timing, power consumption, or electromagnetic radiation) to gain information about the secret key used in the system. The Montgomery ladder algorithm is designed to be constant-time, meaning that it takes the same amount of time to execute regardless of the input, which makes it resistant to these types of attacks.

The Montgomery ladder algorithm operates by iteratively applying two operations to a point on the elliptic curve. First, it adds the point to itself (doubling the point) if the next bit in the scalar is a 0, and second, it adds the point to another point (adding the two points together) if the next bit in the scalar is a 1. The algorithm proceeds in this way until all bits in the scalar have been processed, at which point the final point is returned as the result of the scalar multiplication.

Sliding window algorithm:

The sliding window algorithm is another scalar multiplication algorithm that is used to compute the scalar multiplication of a point on an elliptic curve. Like Montgomery's ladder algorithm, it is also resistant to side-channel attacks.

The sliding window algorithm operates by precomputing a table of points on the elliptic curve that are multiples of a small number of points. This table is then used to speed up the computation of scalar multiplication. Specifically, the scalar is represented as a binary number and is processed in a sliding window of a fixed number of bits. The algorithm then uses this sliding window to compute the scalar multiplication by performing a series of additions and doublings on the points in the precomputed table. By using a sliding window, the algorithm can take advantage of the fact that most bits in the scalar are 0 and skip over these bits in the computation.

Overall, the classical binary extended Euclidean algorithm, Montgomery's ladder algorithm, and the sliding window algorithm are all useful tools in elliptic curve cryptography. The binary extended Euclidean algorithm is used to compute the greatest common divisor of two integers, while Montgomery's ladder algorithm and the sliding window algorithm are used to compute the scalar multiplication of a point on an elliptic curve.

164> could you teach me more about the modular inversion concept?

Certainly! In modular arithmetic, the modular inverse of an integer a with respect to a modulus m is another integer b such that the product of a and b modulo m equals 1. In other words, b is the "multiplicative inverse" of a modulo m.

The modular inverse is useful in many areas of cryptography, such as key generation, digital signatures, and authentication protocols.

There are various algorithms to compute modular inverses, such as the extended Euclidean algorithm, binary extended Euclidean algorithm, Montgomery inversion, and others. These algorithms differ in their performance characteristics and suitability for different scenarios.

For example, the extended Euclidean algorithm is a general-purpose algorithm that can compute modular inverses for any modulus, but it may not be the most efficient algorithm in certain situations. On the other hand, Montgomery inversion is a specialized algorithm that is particularly efficient for modular inverses in prime fields, such as those used in elliptic curve cryptography.

Overall, modular inversion is an important concept in cryptography and efficient algorithms for computing modular inverses are crucial for the performance and security of cryptographic systems.

165> why do we need it in the context of ecdh?

In the context of Elliptic Curve Diffie-Hellman (ECDH), modular inversion is used to compute the shared secret between two parties.

In ECDH, each party generates a private key and a corresponding public key on the chosen elliptic curve. The two parties then exchange their public keys and use them to compute a shared secret.

To compute the shared secret, one party multiplies their private key with the other party's public key, and the other party multiplies their private key with the first party's public key. This results in two points on the elliptic curve, and these points are equal to each other up to a multiple of the generator point (which is publicly known). The shared secret is the x-coordinate of either of these points.

However, computing this shared secret requires modular inversion in the underlying finite field of the elliptic curve. Specifically, the private key of each party is a random integer modulo the order of the generator point, and computing the shared secret involves dividing by this integer modulo the order of the generator point. This is equivalent to computing the modular inverse of the private key modulo the order of the generator point.

Therefore, modular inversion is a crucial step in the computation of the shared secret in ECDH.

166> what are its mathematical properties?

Modular inversion is a mathematical operation performed on two integers, where one integer is divided by another integer in a modular arithmetic system. The result of the operation is another integer that satisfies a congruence relationship with respect to a given modulus.

In general, modular inversion can be represented as:

a⁻¹ ≡ b (mod m)

where a, b, and m are integers, and a and m are coprime (i.e., they have no common divisors other than 1). The modular inverse a⁻¹ is the integer such that a times a⁻¹ is congruent to 1 modulo m, i.e., a × a⁻¹ ≡ 1 (mod m).

Modular inversion has several important properties, including:

1. Unique inverse: If a and m are coprime, then the modular inverse of a modulo m is unique.

2. Associativity: The modular inverse of a product of integers modulo m is equal to the product of the modular inverses of the individual integers modulo m, i.e., (ab)⁻¹ ≡ a⁻¹b⁻¹ (mod m).

3. Distributivity: The modular inverse of a sum of integers modulo m is generally not equal to the sum of the modular inverses of the individual integers modulo m, but the following distributive law holds: (a+b)⁻¹ ≡ a⁻¹ + b⁻¹

   • kab⁻¹ (mod m), where k is an integer such that k(a+b) ≡ 1 (mod m).

Modular inversion is a fundamental operation in elliptic curve cryptography, and is used in many cryptographic protocols and algorithms, including ECDH and ECDSA. In ECDH, modular inversion is used to compute the shared secret key between two parties, while in ECDSA, it is used to compute the signature of a message.

167> what should we watch out for when using this property in code?

When implementing modular inversion in code, it is important to watch out for edge cases and potential errors. Here are some things to keep in mind:

1. Division by zero: Modular inversion requires division by a number which may be zero. In this case, the result is undefined, so it's important to check for this condition and handle it appropriately.

2. Non-invertible elements: Not all elements have a modular inverse. For example, if the modulus and the element have a common factor, then the element will not have an inverse. It is important to check for this condition before attempting to compute the inverse, otherwise the program may crash or return an incorrect result.

3. Numerical precision: When working with large numbers, it is important to ensure that the program has sufficient numerical precision to avoid overflow or underflow errors. For example, if the modulus is very large, then computing the inverse using the extended Euclidean algorithm may require many iterations, which can be computationally expensive and can lead to numerical errors if the intermediate results are not handled carefully.

4. Timing attacks: The time taken to compute the modular inverse can reveal information about the input data, which can be exploited by an attacker. To prevent timing attacks, it is important to use constant-time algorithms that do not depend on the input data, and to avoid branching or other operations that can leak information about the computation.