COUNTING FIXED POINTS, TWO-CYCLES, AND COLLISIONS OF THE DISCRETE EXPONENTIAL FUNCTION USING p-ADIC METHODS

Abstract

Abstract. Brizolis asked for which primes p greater than 3 does there exist a pair (g, h) such that h is a fixed point of the discrete exponential map with base g, or equivalently h is a fixed point of the discrete logarithm with base g. Zhang (1995) and Cobeli and Zaharescu (1999) answered with a "yes" for sufficiently large primes and gave estimates for the number of such pairs when g and h are primitive roots modulo p. In 2000, Campbell showed that the answer to Brizolis was "yes" for all primes. The first author has extended this question to questions about counting fixed points, two-cycles, and collisions of the discrete exponential map. In this paper, we use p-adic methods, primarily Hensel's lemma and p-adic interpolation, to count fixed points, two cycles, collisions, and solutions to related equations modulo powers of a prime p.

Counting Fixed Points and Rooted Closed Walks of the Singular Map $$x \mapsto {x^{{x^n}}}$$ Modulo Powers of a Prime

“…However, we expect that continuing our technique from Section 4 of reduction to one p-adic variable would result in such unwieldy formulas that the solution would not be worthwhile. Unfortunately, the multivariable technique used in [18,Section 5] only appears to apply to the cases that are nonsingular modulo p. A theory of lifting for points that are singular modulo p for multiple equations in multiple variables would be very useful here.…”

“…Two other types of congruences modulo p e involving the self-power map were studied in [18], namely x x ≡ c (mod p e ) and x x ≡ y y (mod p e ). These could also be generalized to the expression x g(x) studied here.…”

“…Its study has accelerated in recent years due to both improvements in technique (see, for instance, [1-3, 7, 8, 11-13, 15-18, 21, 25]) and its relation to a variation of the ElGamal digital signature scheme given in, e.g., [23,Note 11.71]. In particular, [18] and [24] used p-adic techniques to investigate solutions to the equations (among others) (1) x x ≡ c mod p e for fixed c and x in {1, . .…”

Artin's Primitive Root Conjecture – A Survey

“…A preliminary approach to these questions was made by Holden and Robinson [237] using p-adic methods, primarily Hensel's lemma and p-adic interpolation.…”

On the Fixed Points of an Elliptic-Curve Version of Self-Power Map

“…An h is said to be a fixed point of f g if f g ðhÞ ¼ h. For example, if h 2 F Â p generates F Â p and h is also in ðZ=ð p À 1ÞZÞ Â , then we may define f g so that h can be a fixed point by putting g ¼ h h , where hh 1 (mod p À 1Þ. The number of such pairs ðg; hÞ and other properties have been well studied [4][5][6] because too many fixed points could affect the security of cryptosystems based on the difficulty of discrete logarithm problem, which is the problem to compute f À1 g . As a variant of f g , we can define the self-power map modulo a prime: f s ðxÞ ¼ x x mod p. The fixed points of f s have been explored also from a cryptographic viewpoint [1,3].…”