Method for the cryptographic hashing of a message M, characterized in that it comprises the following steps: - forming a sequence (M1,...Mi,...,Mc) of data m-uplets M1 = (a 1,i,...,a 1,m),...,Mi = (a i,1,...,a i,m),...,Mc = (a c,1,...,a c,m), where m is a strictly positive integer, from said message M, - iteratively computing successive output p-uplets V 1,...V i,...,V c, where p is a strictly positive integer, corresponding to said sequence (M1,...Mi,...,Mc) of data m-uplets as a function of at least one set of multivariate polynomials defined over a finite field, said iterative computation of the successive output p-uplets (V 1,...V i,...,V c) comprising the following steps: - defining an initialization input p-uplet E 0 = (e 01,...,e 0p) in the finite field, - applying said initialization input p-uplet E 0 and a first data m-uplet M1 to said at least one set of multivariate polynomials to compute a first output p-uplet V 1 corresponding to said first data m-uplet M1, - defining, on each current new iteration number i, a current input p-uplet E i = (e i1,...,e ip) in the finite field which is a function of at least one of the output p-uplets (V 1,...,V i-1) computed on the preceding iterations, and applying said current input p-uplet E i and a current data m-uplet Mi to said at least one set of multivariate polynomials to compute a current output p-uplet V i corresponding to said current data m-uplet Mi, and - determining a hash value of said message M as a function of the last output p-uplet V c. - forming a sequence (M1,...Mi,...,Mc) of data m-uplets M1 = (a 1,i,...,a 1,m),...,Mi = (a i,1,...,a i,m),...,Mc = (a c,1,...,a c,m), where m is a strictly positive integer, from said message M, - iteratively computing successive output p-uplets V 1,...V i,...,V c, where p is a strictly positive integer, corresponding to said sequence (M1,...Mi,...,Mc) of data m-uplets as a function of at least one set of multivariate polynomials defined over a finite field, said iterative computation of the successive output p-uplets (V 1,...V i,...,V c) comprising the following steps: - defining an initialization input p-uplet E 0 = (e 01,...,e 0p) in the finite field, - applying said initialization input p-uplet E 0 and a first data m-uplet M1 to said at least one set of multivariate polynomials to compute a first output p-uplet V 1 corresponding to said first data m-uplet M1, - defining, on each current new iteration number i, a current input p-uplet E i = (e i1,...,e ip) in the finite field which is a function of at least one of the output p-uplets (V 1,...,V i-1) computed on the preceding iterations, and applying said current input p-uplet E i and a current data m-uplet Mi to said at least one set of multivariate polynomials to compute a current output p-uplet V i corresponding to said current data m-uplet Mi, and - determining a hash value of said message M as a function of the last output p-uplet V c. - defining an initialization input p-uplet E 0 = (e 01,...,e 0p) in the finite field, - applying said initialization input p-uplet E 0 and a first data m-uplet M1 to said at least one set of multivariate polynomials to compute a first output p-uplet V 1 corresponding to said first data m-uplet M1, - defining, on each current new iteration number i, a current input p-uplet E i = (e i1,...,e ip) in the finite field which is a function of at least one of the output p-uplets (V 1,...,V i-1) computed on the preceding iterations, and applying said current input p-uplet E i and a current data m-uplet Mi to said at least one set of multivariate polynomials to compute a current output p-uplet V i corresponding to said current data m-uplet Mi, and - determining a hash value of said message M as a function of the last output p-uplet V c.