Let k be a totally real number field and let k∞ be its cyclotomic Zp-extension, p ≥ 2. This paper synthesizes and generalizes our articles written in french: "Approche p-adique de la conjecture de Greenberg pour les corps totalement réels", Ann. Math. Blaise Pascal 24(2) (2017), 235-291 and "Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg", Ann. math. du Québec 43 (2019), 249-280. We show that this conjecture (nullity of the Iwasawa invariants λ, µ) depends on some images (of ideal norms along the stages kn/k of the tower) in the torsion group Tk of the Galois group of the maximal abelian p-ramified prop extension of k; more precisely these images (obtained, for each fixed n, inductively via a classical algorithm in kn) take place both in the p-class group Clk and in the normalized p-adic regulator Rk of k (recall that #Tk = #Clk #Rk #Wk , where the fixed field of Wk is the Bertrandias-Payan field). A suitable assumption of uniform p-adic distribution of these images, related to Clk and Rk , would constitute a proof of Greenberg's conjecture, which remains hopeless within the sole framework of Iwasawa's theory. We interpret the conjecture in terms of algorithmic complexity, governed by the arithmetic structure of Tk , for which some heuristics and probabilities, and possibly accessible proofs, apply. No assumption is made on the degree of k nor on the decomposition of p in k/Q.