A conditional version of Sibson's $α$-information is defined using a simple closed-form "log-expectation" expression, which satisfies important properties such as consistency, uniform expansion, and data processing inequalities. This definition is compared to previous ones, which in contrast do not satisfy all of these properties. Based on our proposal and on a generalized Fano inequality, we extend the case $α= 1$ of previous works to obtain sharp universal upper bounds for the probability of success of any type side-channel attack, particularly when $α= 2$.