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is necessary to search for smooth numbers (i.e. numbers with small prime factors) of order n1/2. The size of these values is exponential in the size of...

time of ordered and unordered lists and structured trees, and the often-exponential storage requirements of direct access of state spaces of large or variable-length...

moduli. Although dramatically faster than the systematic search, this method also has an exponential time complexity and is therefore not used on computers...

into an equivalent conjunctive normal form, which may, however, be exponentially longer. For example, transforming the formula (x1∧y1) ∨ (x2∧y2) ∨ ....

backtracking search procedure to explore the (exponentially sized) space of variable assignments looking for satisfying assignments. The basic search procedure...

the sum of two or more unknowns, with coefficients, to a constant. An exponential Diophantine equation is one in which unknowns can appear in exponents...

human-oriented tools for guiding the problem resolution. Satisfiability modulo theories for solvers of logical formulas with respect to combinations of...

be defined as primes p such that the multiplicative orders of 2 modulo p and modulo p2 coincide: ordp2 2 = ordp 2, (By the way, ord10932 = 364, and ord35112...

a prime, then the only square roots of 1 modulo n are 1 and −1. Proof Certainly 1 and −1, when squared modulo n, always yield 1. It remains to show that...

, slot[num_slots - 1] function find_slot(key) i := hash(key) modulo num_slots // search until we either find the key, or find an empty slot. while (slot[i]...

cryptanalysis that mix additions over two different groups (e.g. addition modulo 232 and addition over GF(2)) and where input and output differences are...

from more abstract settings (for example, if the coefficients are integers modulo some prime number p, or elements of an arbitrary ring), the formula for...

that often involves DPLL is automated theorem proving or satisfiability modulo theories (SMT), which is a SAT problem in which propositional variables...

Ridder's method: 3-point, exponential scaling Secant method: 2-point, 1-sided Hybrid Algorithms Alpha–beta pruning: search to reduce number of nodes in...

there is a chance to discover this by computing c 5 {\displaystyle c_{5}} modulo some small prime p {\displaystyle p} (using recursive modular exponentiation)...

determined by the value of p modulo 5. If p is congruent to 1 or 4 modulo 5, then p divides Fp−1, and if p is congruent to 2 or 3 modulo 5, then, p divides Fp+1...

GF(p) is the prime field of order p; it is the field of residue classes modulo p, and its p elements are denoted 0, 1, ..., p−1. Thus a = b in GF(p) means...

group of non-zero integers modulo p under multiplication, which has order p − 1. Notice that 2 (strictly speaking, its image modulo p) has multiplicative order...

values. Alternatively, Brent's algorithm is based on the idea of exponential search. Both Floyd's and Brent's algorithms use only a constant number of...

crossings. Therefore, a brute force search for all sequences of Reidemeister moves can detect unknottedness in exponential time. Similarly, any two triangulations...