Volume 31, Issue 1

We construct a probabilistic polynomial algorithm that solves the integer factorization problem using an oracle solving the Diffie–Hellman problem.

Two Latin squares A , B of order n are called pseudo orthogonal if for any 1 ≤ i , j ≤ n there exists a k , 1 ≤ k ≤ n , such that A ( i , k ) = B ( j , k ). We prove that the existence of a family of m mutually pseudo orthogonal Latin squares of order n is equivalent to the existence of a family of m mutually orthogonal Latin squares of order n . We also obtain exact values of clique partition numbers of several classes of complete multipartite graphs and of the tensor product of complete graphs.

We study the threshold probability for the existence of a panchromatic coloring with r colors for a random k -homogeneous hypergraph in the binomial model H ( n , k , p ), that is, a coloring such that each edge of the hypergraph contains the vertices of all r colors. It is shown that this threshold probability for fixed r < k and increasing n corresponds to the sparse case, i. e. the case p=cn/(nk)$p = cn/{n \choose k}$ for positive fixed c . Estimates of the form c 1 ( r , k ) < c < c 2 ( r , k ) for the parameter c are found such that the difference c 2 ( r , k ) − c 1 ( r , k ) converges exponentially fast to zero if r is fixed and k tends to infinity.

We consider configuration graphs with vertex degrees being independent identically distributed random variables. The distribution of these variables satisfies only relatively weak constraints on the probabilities of large values of degrees. For the case when the number of vertices tends to infinity, the conditions are found under which the graph is asymptotically almost surely connected. We also give estimates of the rate of convergence to zero of the probability that the graph is not connected.

Dedicated to the memory of Alexander Ivanovich Pavlov . We consider the set of n -permutations with cycle lengths belonging to some fixed set A of natural numbers (so-called A -permutations). Let random permutation τ n be uniformly distributed on this set. For some class of sets A we find the asymptotics with remainder term for moments of total cycle number of τ n .

The paper is concerned with the complexity and depth of reversible circuits consisting of NOT, CNOT, and 2-CNOT gates under constraints on the number of additional inputs. We study the Shannon functions for the complexity L ( n , q ) and depth D ( n , q ) of a reversible circuit implementing a map f:ℤ2n→ℤ2n$f\colon \mathbb{Z}_2^n \to \mathbb{Z}_2^n$ under the condition that the number of additional inputs q is in the range 8n<q≲n2n−⌈n /ϕ(n)⌉$8n < q \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} n{2^{n - \left\lceil {n{\rm{ }}/\phi (n)} \right\rceil }}$, where ϕ ( n ) → ∞ and n / ϕ ( n ) − log 2 n → ∞ as n → ∞. We establish the upper estimates L(n,q)≲2n+8n2n /(log2(q−4n)−log2n−2)$L(n,q) \lesssim 2^n + 8n2^n \mathop / (\log_2 (q-4n) - \log_2 n - 2)$ and D(n,q)≲2n+1(2,5+log2n−log2(log2(q−4n)−log2n−2))$D(n,q) \lesssim 2^{n+1}(2,5 + \log_2 n - \log_2 (\log_2 (q - 4n) - \log_2 n - 2))$ for this range of q . The asymptotics L(n,q)≍n2n /log2q$L(n,q) \asymp n2^n \mathop / \log_2 q$ is established for q such that n2≲q≲n2n−⌈n /ϕ(n)⌉${n^2} \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} q \mathbin{\lower.3ex\hbox{$\buildrel<\over {\smash{\scriptstyle\sim}\vphantom{_x}}$}} n{2^{n - \left\lceil {n{\rm{ }}/\phi (n)} \right\rceil }}$, where ϕ ( n ) → ∞ and n / ϕ ( n ) − log 2 n → ∞ as n → ∞.