A note on secure multiparty computation via higher residue symbols

We generalize a protocol by Yu for comparing two integers with relatively small difference in a secure multiparty computation setting. Yu's protocol is based on the Legendre symbol. A prime number p is found for which the Legendre symbol (· | p) agrees with the sign function for integers in a certain range {−N, . . . , N} ⊂ ℤ. This can then be computed efficiently. We generalize this idea to higher residue symbols in cyclotomic rings ℤ[ζr ] for r a small odd prime. We present a way to determine a prime number p such that the r-th residue symbol (· | p) r agrees with a desired function f : A → { ζ r 0 , … , ζ r r − 1 } f:A \to \left\{ {\zeta _r^0, \ldots ,\zeta _r^{r - 1}} \right\} on a given small subset A ⊂ ℤ[ζr ], when this is possible. We also explain how to efficiently compute the r-th residue symbol in a secret shared setting.

Using decomposition groups to prove theorems about quadratic residues

In this text we elaborate on the modern viewpoint of the quadratic reciprocity law via methods of alge- braic number theory and class field theory. We present original short and simple proofs of so called addi- tional quadratic reciprocity laws and of the multiplicativity of the Legendre symbol using decompositon groups of primes in quadratic and cyclotomic extensions of Q.

Linear Complexity of Binary Threshold Sequences Derived from Generalized Polynomial Quotient with Prime-Power Modulus

In this paper, firstly we extend the polynomial quotient modulo an odd prime [Formula: see text] to its general case with modulo [Formula: see text] and [Formula: see text]. From the new quotient proposed, we define a class of [Formula: see text]-periodic binary threshold sequences. Then combining the Legendre symbol and Euler quotient modulo [Formula: see text] together, with the condition of [Formula: see text], we present exact values of the linear complexity for [Formula: see text], and all the possible values of the linear complexity for [Formula: see text]. The linear complexity is very close to the period and is of desired value for cryptographic purpose. Our results extend the linear complexity results of the corresponding [Formula: see text]-periodic binary sequences in earlier work.

Quadratic residues and quartic residues modulo primes

In this paper, we study some products related to quadratic residues and quartic residues modulo primes. Let [Formula: see text] be an odd prime and let [Formula: see text] be any integer. We determine completely the product [Formula: see text] modulo [Formula: see text]; for example, if [Formula: see text] then [Formula: see text] where [Formula: see text] denotes the Legendre symbol. We also determine [Formula: see text] modulo [Formula: see text].

Cryptanalysis of the Legendre PRF and Generalizations

The Legendre PRF relies on the conjectured pseudorandomness properties of the Legendre symbol with a hidden shift. Originally proposed as a PRG by Damgård at CRYPTO 1988, it was recently suggested as an efficient PRF for multiparty computation purposes by Grassi et al. at CCS 2016. Moreover, the Legendre PRF is being considered for usage in the Ethereum 2.0 blockchain.This paper improves previous attacks on the Legendre PRF and its higher-degree variant due to Khovratovich by reducing the time complexity from O(< (p log p/M) to O(p log2 p/M2) Legendre symbol evaluations when M ≤ 4√ p log2 p queries are available. The practical relevance of our improved attack is demonstrated by breaking three concrete instances of the PRF proposed by the Ethereum foundation. Furthermore, we generalize our attack in a nontrivial way to the higher-degree variant of the Legendre PRF and we point out a large class of weak keys for this construction. Lastly, we provide the first security analysis of two additional generalizations of the Legendre PRF originally proposed by Damgård in the PRG setting, namely the Jacobi PRF and the power residue PRF.