-----BEGIN PGP SIGNED MESSAGE----- I think that the proposed one-way function is too easy to invert. To decrypt a message requires producing the inverses for each function, which in turn requires knowing the constants (c1, c2, ...). But these constants can be easily obtained from expanding the general form of the equation and setting coefficients equal to one another. The resulting equations are in an easy to solve form (I think it's called upper triagonal form in linear algebra (?)). As it turns out, only a fraction of the simultaneous equations generated are needed to solve for the constants. For instance, at the end of this message is the polynomial obtained from the following nested equations: a = 1/2 x^2 + 1/2 x + c1 b = 1/2 a^2 + 1/2 a + c2 c = 1/2 b^2 + 1/2 b + c3 d = 1/2 c^2 + 1/2 c + c4 d is then the resulting 16th degree polynomial at the end of this message. [By the way, I used Mathematica to do this - the Expand[] function.] The reason the resulting simultaneous equations can be solved so easily is they are in a form convenient for back-substitution: The coefficient for x^14 is (9/8192 + c1/2048) x^12 is (139/16384 + c1 7/512 + 7/2048 c1^2 + c2/1024) etc. The first equation immediately yields c1; this in the next yields c2, etc. So the various constants can be obtained with no trouble. Here is the polynomial d: c1/8 + (7*c1^2)/32 + (7*c1^3)/32 + (25*c1^4)/128 + c1^5/8 + (5*c1^6)/64 + c1^7/32 + c1^8/128 + c2/4 + (3*c1*c2)/8 + (9*c1^2*c2)/16 + (7*c1^3*c2)/16 + (3*c1^4*c2)/8 + (3*c1^5*c2)/16 + (c1^6*c2)/16 + (3*c2^2)/8 + (3*c1*c2^2)/8 + (9*c1^2*c2^2)/16 + (3*c1^3*c2^2)/8 + (3*c1^4*c2^2)/16 + c2^3/4 + (c1*c2^3)/4 + (c1^2*c2^3)/4 + c2^4/8 + c3/2 + (c1*c3)/4 + (3*c1^2*c3)/8 + (c1^3*c3)/4 + (c1^4*c3)/8 + (c2*c3)/2 + (c1*c2*c3)/2 + (c1^2*c2*c3)/2 + (c2^2*c3)/2 + c3^2/2 + c4 + x/16 + (7*c1*x)/32 + (21*c1^2*x)/64 + (25*c1^3*x)/64 + (5*c1^4*x)/16 + (15*c1^5*x)/64 + (7*c1^6*x)/64 + (c1^7*x)/32 + (3*c2*x)/16 + (9*c1*c2*x)/16 + (21*c1^2*c2*x)/32 + (3*c1^3*c2*x)/4 + (15*c1^4*c2*x)/32 + (3*c1^5*c2*x)/16 + (3*c2^2*x)/16 + (9*c1*c2^2*x)/16 + (9*c1^2*c2^2*x)/16 + (3*c1^3*c2^2*x)/8 + (c2^3*x)/8 + (c1*c2^3*x)/4 + (c3*x)/8 + (3*c1*c3*x)/8 + (3*c1^2*c3*x)/8 + (c1^3*c3*x)/4 + (c2*c3*x)/4 + (c1*c2*c3*x)/2 + (15*x^2)/128 + (49*c1*x^2)/128 + (159*c1^2*x^2)/256 + (45*c1^3*x^2)/64 + (155*c1^4*x^2)/256 + (51*c1^5*x^2)/128 + (21*c1^6*x^2)/128 + (c1^7*x^2)/32 + (21*c2*x^2)/64 + (57*c1*c2*x^2)/64 + (39*c1^2*c2*x^2)/32 + (39*c1^3*c2*x^2)/32 + (45*c1^4*c2*x^2)/64 + (3*c1^5*c2*x^2)/16 + (21*c2^2*x^2)/64 + (27*c1*c2^2*x^2)/32 + (27*c1^2*c2^2*x^2)/32 + (3*c1^3*c2^2*x^2)/8 + (3*c2^3*x^2)/16 + (c1*c2^3*x^2)/4 + (7*c3*x^2)/32 + (9*c1*c3*x^2)/16 + (9*c1^2*c3*x^2)/16 + (c1^3*c3*x^2)/4 + (3*c2*c3*x^2)/8 + (c1*c2*c3*x^2)/2 + (35*x^3)/256 + (109*c1*x^3)/256 + (95*c1^2*x^3)/128 + (105*c1^3*x^3)/128 + (185*c1^4*x^3)/256 + (49*c1^5*x^3)/128 + (7*c1^6*x^3)/64 + (43*c2*x^3)/128 + (27*c1*c2*x^3)/32 + (87*c1^2*c2*x^3)/64 + (35*c1^3*c2*x^3)/32 + (15*c1^4*c2*x^3)/32 + (21*c2^2*x^3)/64 + (21*c1*c2^2*x^3)/32 + (9*c1^2*c2^2*x^3)/16 + (c2^3*x^3)/8 + (7*c3*x^3)/32 + (7*c1*c3*x^3)/16 + (3*c1^2*c3*x^3)/8 + (c2*c3*x^3)/4 + (305*x^4)/2048 + (127*c1*x^4)/256 + (855*c1^2*x^4)/1024 + (495*c1^3*x^4)/512 + (755*c1^4*x^4)/1024 + (21*c1^5*x^4)/64 + (7*c1^6*x^4)/128 + (21*c2*x^4)/64 + (243*c1*c2*x^4)/256 + (339*c1^2*c2*x^4)/256 + (15*c1^3*c2*x^4)/16 + (15*c1^4*c2*x^4)/64 + (75*c2^2*x^4)/256 + (9*c1*c2^2*x^4)/16 + (9*c1^2*c2^2*x^4)/32 + (c2^3*x^4)/16 + (25*c3*x^4)/128 + (3*c1*c3*x^4)/8 + (3*c1^2*c3*x^4)/16 + (c2*c3*x^4)/8 + (69*x^5)/512 + (475*c1*x^5)/1024 + (801*c1^2*x^5)/1024 + (447*c1^3*x^5)/512 + (35*c1^4*x^5)/64 + (21*c1^5*x^5)/128 + (135*c2*x^5)/512 + (207*c1*c2*x^5)/256 + (15*c1^2*c2*x^5)/16 + (15*c1^3*c2*x^5)/32 + (3*c2^2*x^5)/16 + (9*c1*c2^2*x^5)/32 + (c3*x^5)/8 + (3*c1*c3*x^5)/16 + (497*x^6)/4096 + (837*c1*x^6)/2048 + (1437*c1^2*x^6)/2048 + (345*c1^3*x^6)/512 + (175*c1^4*x^6)/512 + (7*c1^5*x^6)/128 + (231*c2*x^6)/1024 + (153*c1*c2*x^6)/256 + (75*c1^2*c2*x^6)/128 + (5*c1^3*c2*x^6)/32 + (15*c2^2*x^6)/128 + (3*c1*c2^2*x^6)/32 + (5*c3*x^6)/64 + (c1*c3*x^6)/16 + (391*x^7)/4096 + (663*c1*x^7)/2048 + (531*c1^2*x^7)/1024 + (105*c1^3*x^7)/256 + (35*c1^4*x^7)/256 + (81*c2*x^7)/512 + (45*c1*c2*x^7)/128 + (15*c1^2*c2*x^7)/64 + (3*c2^2*x^7)/64 + (c3*x^7)/32 + (2337*x^8)/32768 + (123*c1*x^8)/512 + (675*c1^2*x^8)/2048 + (105*c1^3*x^8)/512 + (35*c1^4*x^8)/1024 + (99*c2*x^8)/1024 + (45*c1*c2*x^8)/256 + (15*c1^2*c2*x^8)/256 + (3*c2^2*x^8)/256 + (c3*x^8)/128 + (101*x^9)/2048 + (311*c1*x^9)/2048 + (175*c1^2*x^9)/1024 + (35*c1^3*x^9)/512 + (25*c2*x^9)/512 + (15*c1*c2*x^9)/256 + (259*x^10)/8192 + (85*c1*x^10)/1024 + (147*c1^2*x^10)/2048 + (7*c1^3*x^10)/512 + (21*c2*x^10)/1024 + (3*c1*c2*x^10)/256 + (9*x^11)/512 + (77*c1*x^11)/2048 + (21*c1^2*x^11)/1024 + (3*c2*x^11)/512 + (139*x^12)/16384 + (7*c1*x^12)/512 + (7*c1^2*x^12)/2048 + (c2*x^12)/1024 + (7*x^13)/2048 + (7*c1*x^13)/2048 + (9*x^14)/8192 + (c1*x^14)/2048 + x^15/4096 + x^16/32768 -----BEGIN PGP SIGNATURE----- Version: 2.3a iQCVAgUBLMGZfoOA7OpLWtYzAQHZ7QP8Dl0YlZaDCcOmloRmxVH7s3eGaARM6xBx q38k3ck6zw6bCFRxR2rQFflokxauEZ455l8sJv3iMJYTimORoetq6zEygZ8Wchsa 5/P1kZJL4sIQYkMuc/+iZqad9WJZz5nerHRQ/nu+2kfBJCCl8Xrvytwg9xhO4s4G sCUccLBHuIA= =BE17 -----END PGP SIGNATURE-----