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Can Shamir’s secret-sharing protect its secret even when all shares are partially compromised? For instance, repairing Reed-Solomon codewords, when possible, recovers the entire secret in the corresponding Shamir’s secret sharing. Yet, Shamir’s secret sharing mitigates various side-channel threats, depending on where its "secret-sharing polynomial" is evaluated. Although most evaluation places yield secure schemes, none are known explicitly; even techniques to identify them are unknown. Our work initiates research into such classifier constructions and derandomization objectives. In this work, we focus on Shamir’s scheme over prime fields, where every share is required to reconstruct the secret. We investigate the security of these schemes against single-bit probes into shares stored in their native binary representation. Technical analysis is particularly challenging when dealing with Reed-Solomon codewords over prime fields, as observed recently in the code repair literature. Furthermore, ensuring the statistical independence of the leakage from the secret necessitates the elimination of any subtle correlations between them. In this context, we present: 1) An efficient algorithm to classify evaluation places as secure or vulnerable against the least-significant-bit leakage. 2) Modulus choices where the classifier above extends to any single-bit probe per share. 3) Explicit modulus choices and secure evaluation places for them. On the way, we discover new bit-probing attacks on Shamir’s scheme, revealing surprising correlations between the leakage and the secret, leading to vulnerabilities when choosing evaluation places naïvely. Our results rely on new techniques to analyze the security of secret-sharing schemes against side-channel threats. We connect their leakage resilience to the orthogonality of square wave functions, which, in turn, depends on the 2-adic valuation of rational approximations. These techniques, novel to the security analysis of secret sharings, can potentially be of broader interest.