Repeated Squaring

Repeated squaring is an algorithm that computes large exponentiations, like (a^e \mod n), in logarithmic time (O(\log e)) instead of linear time (O(e)). It works by repeatedly squaring the base and multiplying intermediate results based on the binary representation of the exponent. This efficiency is critical for cryptographic operations that must handle exponents with hundreds or thousands of bits.

• Process: The exponent (e) is expressed in binary. For each bit, the current base is squared. If the bit is 1, the result is multiplied by the current base value.
• Example: To compute (3^{13}), where 13 is 1101 in binary, the algorithm performs far fewer multiplications than 12 sequential multiplications.

This algorithm is the workhorse behind major public-key cryptosystems. Its efficiency makes practical the massive modular exponentiations required for security.

• RSA Encryption/Decryption: The core operations (m^e \mod n) (encryption) and (c^d \mod n) (decryption) rely entirely on repeated squaring for performance.
• Diffie-Hellman Key Exchange: Computing the shared secret (g^{ab} \mod p) uses this method.
• Digital Signatures: Schemes like DSA and Schnorr signatures depend on efficient exponentiation in cyclic groups.

In zero-knowledge proof systems, especially zk-SNARKs, repeated squaring is essential for computing polynomial commitments and verifying proofs within elliptic curve groups. The prover and verifier perform many group exponentiation operations, which would be infeasible without the logarithmic speedup.

• Exponentiation in Groups: Operations like (g^a) in a cryptographic group (e.g., BN254, BLS12-381) are fundamental to proof generation and verification.
• Circuit Constraints: Some proof systems express computations as arithmetic circuits where repeated squaring optimizes operations involving large field elements.

Repeated squaring appears in modern consensus mechanisms and cryptographic primitives used in blockchains.

• Verifiable Delay Functions (VDFs): Some VDF constructions, like those based on sequential squaring in a group (e.g., RSA group or class groups), are literally chains of repeated squaring operations. This creates a compute-bound delay that is verifiable much faster.
• Proof-of-Stake Validator Selection: While not always explicit, efficient exponentiation can be involved in cryptographic lotteries or leader election protocols.

Beyond the basic method, several optimizations build upon repeated squaring for even greater performance in production systems.

• Sliding Window Method: Pre-computes powers of the base to reduce the number of multiplications further.
• Montgomery Multiplication: Often paired with repeated squaring to optimize the modular reduction step.
• Fixed-Exponent Exponentiation: For exponents used repeatedly (e.g., RSA public exponent 65537), further specialized optimizations are applied.

Understanding repeated squaring connects to other essential cryptographic and algorithmic primitives.

• Modular Exponentiation: The broader operation that repeated squaring computes efficiently.
• Binary Exponentiation: A synonymous term for the repeated squaring algorithm.
• Discrete Logarithm Problem: The security of many systems using repeated squaring relies on the hardness of this problem.
• Fast Fourier Transform (FFT): Another foundational (O(n \log n)) algorithm; repeated squaring is the analogous optimizer for exponentiation.

Repeated squaring is vulnerable to fault injection attacks, where an adversary induces a computational error (e.g., via voltage glitching) during the exponentiation process. A single corrupted bit in a squaring or multiplication step can produce an erroneous output. By analyzing the faulty result alongside a correct one, an attacker can often deduce the secret exponent. This is a practical threat to hardware security modules (HSMs) and smart cards. Countermeasures involve redundancy checks, such as computing the operation twice and comparing results.

The algorithm's correctness depends on the mathematical properties of modular arithmetic. Key limitations include:

• Modulus size: Extremely large moduli (common in post-quantum cryptography) increase computational load and the risk of overflow or precision errors in some environments.
• Base and modulus relationship: If the base and modulus share a common factor (i.e., are not coprime), it can reveal information about the modulus's factorization, compromising RSA.
• Exponent zero: The algorithm must correctly handle an exponent of 0 to return 1, a simple but critical edge case for protocol correctness.

Writing a correct, efficient, and secure implementation is non-trivial. Common pitfalls that introduce vulnerabilities include:

• Incorrect modular reduction: Failing to reduce modulo n after each squaring and multiplication step can cause integer overflow and incorrect results.
• Using non-constant-time primitives like conditional branches or table lookups based on secret data.
• Lack of input validation, allowing maliciously crafted inputs that cause excessive resource consumption (a denial-of-service vector).
• Inadequate random number generation for exponent blinding, which can itself be predictable.

In zero-knowledge proof systems (e.g., zk-SNARKs), repeated squaring is used within large finite fields. The security and performance limitations here are distinct:

• Prover time: The sequential nature of repeated squaring for large exponents can be a bottleneck for proof generation, limiting throughput.
• Circuit complexity: In arithmetic circuits, exponentiation is represented as many multiplication gates, increasing proof size and verification cost.
• Field-specific attacks: The security depends on the hardness of the Discrete Log Problem in the chosen elliptic curve group or field. Weak parameter selection can lead to vulnerabilities.

Repeated squaring is the standard algorithm for modular exponentiation, computing expressions like (a^b \mod n) efficiently. It reduces the number of multiplications from (O(b)) to (O(\log b)) by exploiting the binary representation of the exponent.

• Process: Square the base repeatedly and multiply intermediate results based on the exponent's binary bits.
• Example: To compute (3^{13} \mod 7), the algorithm processes the binary 1101 for 13, performing far fewer than 13 multiplications.

The RSA cryptosystem relies on repeated squaring for its core operations. Encryption ((c = m^e \mod n)) and decryption ((m = c^d \mod n)) both involve raising a number to a large power modulo (n).

• Key Size: With exponents ((e, d)) often 2048+ bits, naive exponentiation is impossible. Repeated squaring makes these operations feasible in milliseconds.
• Security Foundation: The computational difficulty of reversing this operation without the private key underpins RSA's security.

The Diffie-Hellman protocol uses repeated squaring to compute the shared secret. Each party calculates (g^{ab} \mod p) from public values (g^a) and (g^b), where (a) and (b) are private keys.

• Core Calculation: The final shared secret ((g^a)^b \mod p = g^{ab} \mod p) is computed via repeated squaring.
• Efficiency Critical: This must be fast for real-time key establishment in protocols like TLS and SSH.

Repeated squaring is also known as binary exponentiation or exponentiation by squaring. It's a general algorithm for fast exponentiation, not limited to modular arithmetic.

• Algorithm: Iterate through the bits of the exponent. For each bit, square the current result; if the bit is 1, multiply by the base.
• Complexity: Achieves (O(\log n)) time complexity, making it exponentially faster than the naive linear approach for large numbers.

Repeated squaring enables efficient calculation of modular inverses when the modulus is prime. Using Fermat's Little Theorem, the inverse of (a) modulo prime (p) is (a^{p-2} \mod p).

• Application: This is used in cryptographic schemes to compute private key components and in elliptic curve operations.
• Process: Computing (a^{p-2} \mod p) directly would be prohibitive; repeated squaring makes it practical.

This probabilistic primality test uses repeated squaring in its core routine. It checks if a number is prime by testing congruences of the form (a^{d \cdot 2^r} \mod n).

• Key Step: The test requires computing a sequence of squares modulo (n) starting from (a^d \mod n).
• Cryptographic Use: Essential for generating the large primes needed for RSA keys. The efficiency of repeated squaring allows for fast generation.