WARNING: OLD DRAFT. Define f(n) to be the integer complexity of n, the smallest number of ones needed to write n using an arbitrary combination of addition and multiplication. John Selfridge showed that f(n) ≥ 3 log3 n for all n. Define d(n) = f(n) − 3 log3 n; in this paper we present a method for classifying all n with d(n) < r for a given r. From this, we derive a number of consequences. We prove that f(2m3k) = 2m + 3k for m ≤ 31 with m and k not both zero, and present a method that can, with more computation, prove the same for larger m. We extend a result of Daniel Rawsthorne by finding formulae for the r’th largest number with complexity k, so long as k is sufficiently large relative to r. Defining Ar to be the set of all n with d(n) < r, we prove that Ar(x) = Θ((log x)brc+1). Finally we prove that the set of all values of d is a well-ordered subset of R, with order type ωω , as was conjectured earlier by Juan Arias de Reyna. We also make some conjectures that suggest a possible approach for proving f(n) 3 log3 n.

/pdf/numbers-with-integer-complexity-close-to-the-lower-bound-2xsygktekg.pdf

Numbers with Integer Complexity Close to the Lower Bound

We present three algorithms to compute the complexity $\Vert n\Vert$ of all natural numbers $ n\le N$. The first of them is a brute force algorithm, computing all these complexities in time $O(N^2)$ and space $O(N\log^2 N)$. The main problem of this algorithm is the time needed for the computation. In 2008 there appeared three independent solutions to this problem: V. V. Srinivas and B. R. Shankar [11], M. N. Fuller [7], and J. Arias de Reyna and J. van de Lune [3]. All three are very similar. Only [11] gives an estimation of the performance of its algorithm, proving that the algorithm computes the complexities in time $O(N^{1+\beta})$, where $1+\beta =\log3/\log2\approx1.584963$. The other two algorithms, presented in [7] and [3], were very similar but both superior to the one in [11]. In Section 2 we present a version of these algorithms and in Section 4 it is shown that they run in time $O(N^\alpha)$ and space $O(N\log\log N)$. (Here $\alpha = 1.230175$). 
In Section 2 we present the algorithm of [7] and [3]. The main advantage of this algorithm with respect to that in [11] is the definition of kMax in Section 2.7. This explains the difference in performance from $O(N^{1+\beta})$ to $O(N^\alpha)$. 
In Section 3 we present a detailed description a space-improved algorithm of Fuller and in Section 5 we prove that it runs in time $O(N^\alpha)$ and space $O(N^{(1+\beta)/2}\log\log N)$, where $\alpha=1.230175$ and $(1+\beta)/2\approx0.792481$.

Algorithms for determining integer complexity

Side-channel attacks are the most efficient attacks against cryptosystems. While the classical blackbox attacks only exploit the inputs and outputs of cryptographic algorithms, side-channel attacks also get use of the physical leakage released by the underlying device during algorithms executions. In this thesis, we focus on one kind of side-channel attacks which exploits the power consumption of the underlying device to recover the algorithms secret keys. They are gathered under the term power-analysis attacks. Most of the existing power-analysis attacks rely on the observations of variables which only depend on a few secret bits using a divide-and-conquer strategy. In this thesis, we exhibit new kinds of attacks which exploit the observation of intermediate variables highly dependent on huge secrets. In particular, we show how to recover a 128-bit key by only recording the leakage of the Galois multiplication’s results between several known messages and this secret key. We also study two commonly used algorithmic countermeasures against side-channel attacks: leakage resilience and masking. On the one hand, we define a leakage-resilient encryption scheme based on a regular update of the secret key and we prove its security. On the other hand, we build, using formal methods, a tool to automatically verify the security of masked algorithms. We also exhibit new security and compositional properties which can be used to generate masked algorithms at any security order from their unprotected versions. Finally, we propose a comparison between these two countermeasures in order to help industrial experts to determine the best protection to integrate in their products, according to their constraints in terms of security and performances.

https://hal.inria.fr/tel-01235207/document

Security of cryptosystems against power-analysis attacks

In this dissertation we consider two notions of the “complexity” of a natural number, the first being addition chain length, and the second known simply as “integer complexity”. The integer complexity of n, denoted ‖n‖, is the smallest number of 1’s needed to write n using an arbitrary combination of addition and multiplication. It is known that ‖n‖ ≥ 3 log3 n for all n. We consider the difference δ(n) := ‖n‖ − 3 log3 n, which we call the defect of n. We consider the set of all defects – the set D := {δ(n) : n ∈ N}. We show that, as a subset of the real numbers, D is well-ordered, with order type ω; we also show the same for several variants of this set. Moreover, we show that, for k ≥ 1 a natural number, D ∩ [0, k) has order type precisely ω. We also use the defect to prove stabilization results about ‖n‖. Specifically, for any n, there exists K = K(n) such that for k ≥ K, we have δ(3n) = δ(3n). We call K(n) the stabilization length of n. Finally, we provide a way of, given r > 0, computing all numbers n with δ(n) < r. We use this to show that the stabilization length K(n) is effectively computable. The algorithm is also, empirically, much faster than existing methods for computing ‖2‖, and we use it to prove that ‖23‖ = 2k + 3` for 0 ≤ k ≤ 48 and ` ≥ 0, with k and ` not both 0. In parallel to our results for integer complexity, we also consider addition chain length. An addition chain for n is defined to be a sequence (a0, a1, . . . , ar) such that a0 = 1, ar = n, and, for any k with 1 ≤ k ≤ r, there exist 0 ≤ i, j < k such that ak = ai + aj; the number r is called the length of the addition chain. The shortest

/pdf/integer-complexity-addition-chains-and-well-ordering-7fan1cgjw1.pdf

Integer Complexity, Addition Chains, and Well-Ordering.

Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. Define $n$ to be stable if for all $k\ge 0$, we have $\|3^k n\|=\|n\|+3k$. In [7], this author and Zelinsky showed that for any $n$, there exists some $K=K(n)$ such that $3^K n$ is stable; however, the proof there provided no upper bound on $K(n)$ or any way of computing it. In this paper, we describe an algorithm for computing $K(n)$, and thereby also show that the set of stable numbers is a computable set. The algorithm is based on considering the defect of a number, defined by $\delta(n):=\|n\|-3\log_3 n$, building on the methods presented in [3]. As a side benefit, this algorithm also happens to allow fast evaluation of the complexities of powers of $2$; we use it to verify that $\|2^k 3^\ell\|=2k+3\ell$ for $k\le48$ and arbitrary $\ell$ (excluding the case $k=\ell=0$), providing more evidence for the conjecture that $\|2^k 3^\ell\|=2k+3\ell$ whenever $k$ and $\ell$ are not both zero. An implementation of these algorithms in Haskell is available.

Integer Complexity: Algorithms and Computational Results.

—The integer complexity of a positive integer n, denoted f (n), is defined as the least number of 1's required to represent n, using only 1's, the addition and multiplication operators, and the parentheses. The running time of the algorithm currently used to compute f (n) is Θ(n 2). In this paper we present an algorithm with Θ(n log 2 3) as its running time. We also present a proof of the theorem: the largest solutions of f (m) = 3k, 3k±1 are, respectively, m = 3 k , 3 k ± 3 k−1 .

/pdf/integer-complexity-breaking-the-th-n2-barrier-456u2787qc.pdf

Integer Complexity: Breaking the θ(n2) barrier

/pdf/practical-provable-security-against-side-channel-attacks-2im1paqeza.pdf

Srinivas Vivek Venkatesh

Papers

Integer Complexity: Breaking the θ(n2) barrier

Practical Provable Security against Side-Channel Attacks