What question did this study set out to answer?

To examine the distribution of powers modulo a prime and its implications for RSA security.

May 7, 2026Journal of Mathematical Cryptology0 citationsOpen Access

Distribution of powers modulo p and security of RSA

Puntos clave

To examine the distribution of powers modulo a prime and its implications for RSA security.
Analyzed powers modulo p with respect to a fixed integer a and prime p.
Defined a function G(x) to quantify residues under specific conditions.
Demonstrated G(x) has a specific distribution formula.
Established that security risks in RSA occur when using small exponents, allowing vulnerabilities.

Resumen

Abstract Let p be a prime and a be any fixed positive integer such that gcd (a, φ (p) ) = 1. For 0 G (x) = # m ∈ Z p *: m − (m a mod p) x, G (x) =\#\m {Z₏^{}: m- (m^a mod p) G (x) = 2 x − x 2 p − 1 + O p 1 / 2 ⁡ log 2 ⁡ p. G (x) =2x-{x^2p^-1+O (p^1/2log^2p). This distribution result has an immediate cryptographic consequence. For RSA having public key N, e (N, e) with small exponent e (such as 3 or 65537), we show that there exist at least Ω N 3 / 4 ⁡ log 3

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