## What is/are Modulo Prime?

Modulo Prime - The use of the last difficult problem provides increased security of the signature protocols due to reducing the probability of the potential breaking the protocols, which is connected with potential appearance of the breakthrough solutions of the following two computationally difficult problems: (1) finding discrete logarithm modulo prime and (2) factoring composite number n containing two unknown prime divisors.^{[1]}We develop exact formulas for the distribution of quadratic residues and non-residues in sets of the form a + X = { ( a + x ) mod n ∣ x ∈ X } , where n is a prime or the product of two primes and X is a subset of integers with given Jacobi symbols modulo prime factors of n.

^{[2]}Coding strategies based on QGCs are introduced for three fundamental multi-terminal problems: the Körner-Marton problem for modulo prime-power sums, computation over the multiple access channel (MAC), and MAC with distributed states.

^{[3]}This lower bound depends on the Mordell– Weil rank of A and the reduction of p-torsion points modulo primes above p.

^{[4]}This generalizes the earlier results of Munshi and Holowinsky--Nelson, corresponding to multiplicative Dirichlet characters, and applies in particular to trace functions of small conductor modulo primes.

^{[5]}The algorithm of fast calculation proposed by us in the article (acceleration is more than three times as compared with the Harvey algorithm) of Bernoulli numbers modulo prime numbers can also be successfully used to find irregular prime numbers, irregular pairs (p, n), and also when calculating Iwasawa invariants.

^{[6]}Our new algorithm actually solves its counting version modulo prime p>t, by manipulating generating functions using FFT.

^{[7]}We identify two faults in a published algorithm for fast computation of multiplicative inverses modulo prime powers.

^{[8]}We can ask the same question modulo prime-powers pk.

^{[9]}However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues.

^{[10]}The notion of Hadamard modulo prime (HMP) matrix inherits in basics that of classical real Hadamard matrix.

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## modulo prime power

We identify two faults in a published algorithm for fast computation of multiplicative inverses modulo prime powers.^{[1]}However, modulo prime powers, factoring gets harder because there is non-unique factorization and a combinatorial blowup ensues.

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